E16_Eigen decompasition manual based on matlab.ipynb

{
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  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "8915dcb6",
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from sklearn.preprocessing import Normalizer\n",
    "from sklearn.preprocessing import MinMaxScaler\n",
    "from sklearn.preprocessing import RobustScaler\n",
    "from sklearn.preprocessing import StandardScaler\n",
    "from sklearn.preprocessing import PowerTransformer\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "727a2244",
   "metadata": {},
   "source": [
    "## The Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "4b891951",
   "metadata": {},
   "outputs": [],
   "source": [
    "train_normal = pd.read_csv('./data/normal_training.csv')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "901f5c9e",
   "metadata": {},
   "outputs": [],
   "source": [
    "test_faulty = pd.read_csv('./data/faulty_testing.csv')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b811b0f8",
   "metadata": {},
   "outputs": [],
   "source": [
    "# data preparing\n",
    "df_train = train_normal[(train_normal.simulationRun>=1)].iloc[:500,3:]\n",
    "\n",
    "df_test = train_normal[(train_normal.simulationRun>5)&(train_normal.simulationRun<10)].iloc[:,3:].reset_index(drop=True)\n",
    "# IDV = 1\n",
    "# df_test = test_faulty[(test_faulty.faultNumber==IDV) & \n",
    "#                                (test_faulty.simulationRun==1)].iloc[:,3:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "5dd02375",
   "metadata": {},
   "outputs": [],
   "source": [
    "def scl_pick(name, data):\n",
    "    \"\"\"\n",
    "    the several option for name:\n",
    "    -std\n",
    "    -minmax\n",
    "    -robust\n",
    "    -norm\n",
    "    -pwr\n",
    "    \"\"\"\n",
    "    if name == 'std':\n",
    "        std_scl = StandardScaler().fit(data)\n",
    "        df_scl = std_scl.transform(data)\n",
    "        df_scl = pd.DataFrame(df_scl)\n",
    "    elif name == 'minmax':\n",
    "        MM_scl = MinMaxScaler().fit(data)\n",
    "        df_scl = MM_scl.transform(data)\n",
    "        df_scl = pd.DataFrame(df_scl)\n",
    "    elif name == 'robust':\n",
    "        rb_scl = RobustScaler().fit(data)\n",
    "        df_scl = rb_scl.transform(data)\n",
    "        df_scl = pd.DataFrame(df_scl)\n",
    "    elif name == 'norm': ##### xxxxxx\n",
    "        df_norm = Normalizer().fit(data)\n",
    "        df_scl = df_norm.transform(data)\n",
    "        df_scl = pd.DataFrame(df_scl)\n",
    "    else:\n",
    "        pt_scl = PowerTransformer().fit(data)\n",
    "        df_scl = pt_scl.transform(data)\n",
    "        df_scl = pd.DataFrame(df_scl)\n",
    "    return df_scl"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "dc9ae4f8",
   "metadata": {},
   "outputs": [],
   "source": [
    "#-std-minmax-robust-norm-pwr\n",
    "df_scl = scl_pick('std', df_train) #robust\n",
    "df_scl = round(df_scl, 4)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "63f8a165",
   "metadata": {},
   "source": [
    "## Training Centraling Kernel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "b76650ec",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\m477a\\AppData\\Local\\Temp/ipykernel_14368/409773423.py:3: FutureWarning: Calling a ufunc on non-aligned DataFrames (or DataFrame/Series combination). Currently, the indices are ignored and the result takes the index/columns of the first DataFrame. In the future , the DataFrames/Series will be aligned before applying the ufunc.\n",
      "Convert one of the arguments to a NumPy array (eg 'ufunc(df1, np.asarray(df2)') to keep the current behaviour, or align manually (eg 'df1, df2 = df1.align(df2)') before passing to the ufunc to obtain the future behaviour and silence this warning.\n",
      "  K = np.matmul(A, A.T)\n"
     ]
    }
   ],
   "source": [
    "# training proccess\n",
    "A = df_scl\n",
    "K = np.matmul(A, A.T)\n",
    "K = round(K, 4)\n",
    "K = pd.DataFrame(K)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "c0b2e78b",
   "metadata": {},
   "outputs": [
    {
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       "      <th></th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
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       "      <th>6</th>\n",
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       "      <th>9</th>\n",
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       "      <th>490</th>\n",
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       "      <th>499</th>\n",
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       "  </thead>\n",
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       "      <th>0</th>\n",
       "      <td>11.2548</td>\n",
       "      <td>-1.0064</td>\n",
       "      <td>0.0676</td>\n",
       "      <td>-2.9111</td>\n",
       "      <td>7.1808</td>\n",
       "      <td>2.8011</td>\n",
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       "      <td>0.4064</td>\n",
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       "      <td>3.8501</td>\n",
       "      <td>-9.1363</td>\n",
       "      <td>4.2920</td>\n",
       "      <td>-3.3997</td>\n",
       "      <td>-10.6721</td>\n",
       "      <td>-3.1037</td>\n",
       "      <td>5.0832</td>\n",
       "      <td>-1.1396</td>\n",
       "      <td>-2.0656</td>\n",
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       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-1.0064</td>\n",
       "      <td>15.6484</td>\n",
       "      <td>-1.6438</td>\n",
       "      <td>5.6559</td>\n",
       "      <td>2.5905</td>\n",
       "      <td>-0.4405</td>\n",
       "      <td>6.0025</td>\n",
       "      <td>4.6070</td>\n",
       "      <td>-2.4341</td>\n",
       "      <td>2.6041</td>\n",
       "      <td>...</td>\n",
       "      <td>0.9103</td>\n",
       "      <td>1.0259</td>\n",
       "      <td>-2.7118</td>\n",
       "      <td>0.1543</td>\n",
       "      <td>0.3193</td>\n",
       "      <td>8.1392</td>\n",
       "      <td>5.3727</td>\n",
       "      <td>-5.6366</td>\n",
       "      <td>-8.4340</td>\n",
       "      <td>-1.2429</td>\n",
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       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.0676</td>\n",
       "      <td>-1.6438</td>\n",
       "      <td>26.7534</td>\n",
       "      <td>17.8788</td>\n",
       "      <td>-0.3142</td>\n",
       "      <td>5.1253</td>\n",
       "      <td>3.0534</td>\n",
       "      <td>9.6957</td>\n",
       "      <td>-6.1013</td>\n",
       "      <td>3.9280</td>\n",
       "      <td>...</td>\n",
       "      <td>6.0355</td>\n",
       "      <td>5.8826</td>\n",
       "      <td>9.4580</td>\n",
       "      <td>-6.5921</td>\n",
       "      <td>5.7674</td>\n",
       "      <td>-1.7370</td>\n",
       "      <td>-1.2486</td>\n",
       "      <td>-3.1192</td>\n",
       "      <td>-1.1443</td>\n",
       "      <td>-3.6522</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-2.9111</td>\n",
       "      <td>5.6559</td>\n",
       "      <td>17.8788</td>\n",
       "      <td>28.0760</td>\n",
       "      <td>-1.2178</td>\n",
       "      <td>1.0283</td>\n",
       "      <td>7.3694</td>\n",
       "      <td>7.8789</td>\n",
       "      <td>-6.3967</td>\n",
       "      <td>-0.0438</td>\n",
       "      <td>...</td>\n",
       "      <td>4.3933</td>\n",
       "      <td>5.6325</td>\n",
       "      <td>7.7206</td>\n",
       "      <td>0.8120</td>\n",
       "      <td>2.4678</td>\n",
       "      <td>7.4409</td>\n",
       "      <td>0.0876</td>\n",
       "      <td>-8.0283</td>\n",
       "      <td>-7.5141</td>\n",
       "      <td>1.3204</td>\n",
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       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>7.1808</td>\n",
       "      <td>2.5905</td>\n",
       "      <td>-0.3142</td>\n",
       "      <td>-1.2178</td>\n",
       "      <td>40.4630</td>\n",
       "      <td>34.7837</td>\n",
       "      <td>-1.6995</td>\n",
       "      <td>-4.6649</td>\n",
       "      <td>5.3079</td>\n",
       "      <td>4.0270</td>\n",
       "      <td>...</td>\n",
       "      <td>2.6494</td>\n",
       "      <td>6.9232</td>\n",
       "      <td>-7.3993</td>\n",
       "      <td>0.6237</td>\n",
       "      <td>1.2561</td>\n",
       "      <td>-2.8090</td>\n",
       "      <td>-0.0294</td>\n",
       "      <td>7.9712</td>\n",
       "      <td>-3.6729</td>\n",
       "      <td>2.6613</td>\n",
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       "<p>5 rows × 500 columns</p>\n",
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       "       0        1        2        3        4        5       6       7    \\\n",
       "0  11.2548  -1.0064   0.0676  -2.9111   7.1808   2.8011  0.0056  1.4509   \n",
       "1  -1.0064  15.6484  -1.6438   5.6559   2.5905  -0.4405  6.0025  4.6070   \n",
       "2   0.0676  -1.6438  26.7534  17.8788  -0.3142   5.1253  3.0534  9.6957   \n",
       "3  -2.9111   5.6559  17.8788  28.0760  -1.2178   1.0283  7.3694  7.8789   \n",
       "4   7.1808   2.5905  -0.3142  -1.2178  40.4630  34.7837 -1.6995 -4.6649   \n",
       "\n",
       "      8       9    ...     490     491     492     493     494      495  \\\n",
       "0  0.4064  1.5799  ...  0.5931  3.8501 -9.1363  4.2920 -3.3997 -10.6721   \n",
       "1 -2.4341  2.6041  ...  0.9103  1.0259 -2.7118  0.1543  0.3193   8.1392   \n",
       "2 -6.1013  3.9280  ...  6.0355  5.8826  9.4580 -6.5921  5.7674  -1.7370   \n",
       "3 -6.3967 -0.0438  ...  4.3933  5.6325  7.7206  0.8120  2.4678   7.4409   \n",
       "4  5.3079  4.0270  ...  2.6494  6.9232 -7.3993  0.6237  1.2561  -2.8090   \n",
       "\n",
       "      496     497     498     499  \n",
       "0 -3.1037  5.0832 -1.1396 -2.0656  \n",
       "1  5.3727 -5.6366 -8.4340 -1.2429  \n",
       "2 -1.2486 -3.1192 -1.1443 -3.6522  \n",
       "3  0.0876 -8.0283 -7.5141  1.3204  \n",
       "4 -0.0294  7.9712 -3.6729  2.6613  \n",
       "\n",
       "[5 rows x 500 columns]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "f9ed4b55",
   "metadata": {},
   "outputs": [],
   "source": [
    "s = (len(df_scl),len(df_scl))\n",
    "unit = np.ones(s)/len(df_scl)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "71aac288",
   "metadata": {},
   "outputs": [],
   "source": [
    "#######\n",
    "unit = pd.DataFrame(unit)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "47a5b428",
   "metadata": {},
   "outputs": [],
   "source": [
    "K_c = K-unit@K-K@unit+unit@K@unit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "5b8f2dc2",
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       "      <th>0</th>\n",
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       "      <td>2.801094</td>\n",
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       "      <td>1.579916</td>\n",
       "      <td>...</td>\n",
       "      <td>0.593088</td>\n",
       "      <td>3.850108</td>\n",
       "      <td>-9.136295</td>\n",
       "      <td>4.292013</td>\n",
       "      <td>-3.399712</td>\n",
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       "      <td>-3.103700</td>\n",
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       "      <td>15.648399</td>\n",
       "      <td>-1.643795</td>\n",
       "      <td>5.655894</td>\n",
       "      <td>2.590487</td>\n",
       "      <td>-0.440508</td>\n",
       "      <td>6.002499</td>\n",
       "      <td>4.607007</td>\n",
       "      <td>-2.434111</td>\n",
       "      <td>2.604113</td>\n",
       "      <td>...</td>\n",
       "      <td>0.910285</td>\n",
       "      <td>1.025905</td>\n",
       "      <td>-2.711797</td>\n",
       "      <td>0.154310</td>\n",
       "      <td>0.319285</td>\n",
       "      <td>8.139194</td>\n",
       "      <td>5.372698</td>\n",
       "      <td>-5.636598</td>\n",
       "      <td>-8.434016</td>\n",
       "      <td>-1.242903</td>\n",
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       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.067607</td>\n",
       "      <td>-1.643795</td>\n",
       "      <td>26.753411</td>\n",
       "      <td>17.878799</td>\n",
       "      <td>-0.314207</td>\n",
       "      <td>5.125297</td>\n",
       "      <td>3.053404</td>\n",
       "      <td>9.695713</td>\n",
       "      <td>-6.101305</td>\n",
       "      <td>3.928019</td>\n",
       "      <td>...</td>\n",
       "      <td>6.035491</td>\n",
       "      <td>5.882611</td>\n",
       "      <td>9.458008</td>\n",
       "      <td>-6.592084</td>\n",
       "      <td>5.767391</td>\n",
       "      <td>-1.737001</td>\n",
       "      <td>-1.248596</td>\n",
       "      <td>-3.119192</td>\n",
       "      <td>-1.144310</td>\n",
       "      <td>-3.652198</td>\n",
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       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-2.911104</td>\n",
       "      <td>5.655894</td>\n",
       "      <td>17.878799</td>\n",
       "      <td>28.075988</td>\n",
       "      <td>-1.217818</td>\n",
       "      <td>1.028286</td>\n",
       "      <td>7.369393</td>\n",
       "      <td>7.878902</td>\n",
       "      <td>-6.396716</td>\n",
       "      <td>-0.043792</td>\n",
       "      <td>...</td>\n",
       "      <td>4.393280</td>\n",
       "      <td>5.632500</td>\n",
       "      <td>7.720597</td>\n",
       "      <td>0.812005</td>\n",
       "      <td>2.467780</td>\n",
       "      <td>7.440888</td>\n",
       "      <td>0.087593</td>\n",
       "      <td>-8.028303</td>\n",
       "      <td>-7.514122</td>\n",
       "      <td>1.320391</td>\n",
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       "      <th>4</th>\n",
       "      <td>7.180790</td>\n",
       "      <td>2.590487</td>\n",
       "      <td>-0.314207</td>\n",
       "      <td>-1.217818</td>\n",
       "      <td>40.462976</td>\n",
       "      <td>34.783680</td>\n",
       "      <td>-1.699513</td>\n",
       "      <td>-4.664904</td>\n",
       "      <td>5.307878</td>\n",
       "      <td>4.027002</td>\n",
       "      <td>...</td>\n",
       "      <td>2.649374</td>\n",
       "      <td>6.923194</td>\n",
       "      <td>-7.399309</td>\n",
       "      <td>0.623699</td>\n",
       "      <td>1.256074</td>\n",
       "      <td>-2.809018</td>\n",
       "      <td>-0.029414</td>\n",
       "      <td>7.971191</td>\n",
       "      <td>-3.672928</td>\n",
       "      <td>2.661285</td>\n",
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       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>495</th>\n",
       "      <td>-10.672104</td>\n",
       "      <td>8.139194</td>\n",
       "      <td>-1.737001</td>\n",
       "      <td>7.440888</td>\n",
       "      <td>-2.809018</td>\n",
       "      <td>-4.698614</td>\n",
       "      <td>1.355193</td>\n",
       "      <td>-0.148798</td>\n",
       "      <td>3.515184</td>\n",
       "      <td>8.046708</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.764420</td>\n",
       "      <td>-4.210900</td>\n",
       "      <td>3.462097</td>\n",
       "      <td>-6.719095</td>\n",
       "      <td>18.034280</td>\n",
       "      <td>40.463588</td>\n",
       "      <td>7.860093</td>\n",
       "      <td>-8.158003</td>\n",
       "      <td>4.955378</td>\n",
       "      <td>8.036091</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>496</th>\n",
       "      <td>-3.103700</td>\n",
       "      <td>5.372698</td>\n",
       "      <td>-1.248596</td>\n",
       "      <td>0.087593</td>\n",
       "      <td>-0.029414</td>\n",
       "      <td>-1.633409</td>\n",
       "      <td>-1.137002</td>\n",
       "      <td>0.502606</td>\n",
       "      <td>-8.135812</td>\n",
       "      <td>-5.761088</td>\n",
       "      <td>...</td>\n",
       "      <td>2.460984</td>\n",
       "      <td>2.056504</td>\n",
       "      <td>3.404502</td>\n",
       "      <td>-1.032691</td>\n",
       "      <td>-0.650716</td>\n",
       "      <td>7.860093</td>\n",
       "      <td>29.333497</td>\n",
       "      <td>19.186501</td>\n",
       "      <td>0.829183</td>\n",
       "      <td>-3.456904</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>497</th>\n",
       "      <td>5.083205</td>\n",
       "      <td>-5.636598</td>\n",
       "      <td>-3.119192</td>\n",
       "      <td>-8.028303</td>\n",
       "      <td>7.971191</td>\n",
       "      <td>2.713695</td>\n",
       "      <td>-6.155798</td>\n",
       "      <td>-6.842089</td>\n",
       "      <td>-4.205207</td>\n",
       "      <td>-12.150483</td>\n",
       "      <td>...</td>\n",
       "      <td>-2.619311</td>\n",
       "      <td>4.996009</td>\n",
       "      <td>6.710706</td>\n",
       "      <td>11.294114</td>\n",
       "      <td>-6.044211</td>\n",
       "      <td>-8.158003</td>\n",
       "      <td>19.186501</td>\n",
       "      <td>36.603206</td>\n",
       "      <td>9.409287</td>\n",
       "      <td>3.965700</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>498</th>\n",
       "      <td>-1.139614</td>\n",
       "      <td>-8.434016</td>\n",
       "      <td>-1.144310</td>\n",
       "      <td>-7.514122</td>\n",
       "      <td>-3.672928</td>\n",
       "      <td>-1.294024</td>\n",
       "      <td>-4.383217</td>\n",
       "      <td>-4.969508</td>\n",
       "      <td>2.194474</td>\n",
       "      <td>-4.847502</td>\n",
       "      <td>...</td>\n",
       "      <td>-4.139230</td>\n",
       "      <td>-8.862910</td>\n",
       "      <td>6.183287</td>\n",
       "      <td>-2.552405</td>\n",
       "      <td>6.351270</td>\n",
       "      <td>4.955378</td>\n",
       "      <td>0.829183</td>\n",
       "      <td>9.409287</td>\n",
       "      <td>33.992669</td>\n",
       "      <td>24.624281</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>499</th>\n",
       "      <td>-2.065601</td>\n",
       "      <td>-1.242903</td>\n",
       "      <td>-3.652198</td>\n",
       "      <td>1.320391</td>\n",
       "      <td>2.661285</td>\n",
       "      <td>-1.027211</td>\n",
       "      <td>-6.002004</td>\n",
       "      <td>-8.532495</td>\n",
       "      <td>-2.822213</td>\n",
       "      <td>-5.130689</td>\n",
       "      <td>...</td>\n",
       "      <td>-2.928217</td>\n",
       "      <td>-2.104897</td>\n",
       "      <td>-0.786400</td>\n",
       "      <td>1.265808</td>\n",
       "      <td>1.119183</td>\n",
       "      <td>8.036091</td>\n",
       "      <td>-3.456904</td>\n",
       "      <td>3.965700</td>\n",
       "      <td>24.624281</td>\n",
       "      <td>51.905994</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>500 rows × 500 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "           0          1          2          3          4          5    \\\n",
       "0    11.254804  -1.006399   0.067607  -2.911104   7.180790   2.801094   \n",
       "1    -1.006399  15.648399  -1.643795   5.655894   2.590487  -0.440508   \n",
       "2     0.067607  -1.643795  26.753411  17.878799  -0.314207   5.125297   \n",
       "3    -2.911104   5.655894  17.878799  28.075988  -1.217818   1.028286   \n",
       "4     7.180790   2.590487  -0.314207  -1.217818  40.462976  34.783680   \n",
       "..         ...        ...        ...        ...        ...        ...   \n",
       "495 -10.672104   8.139194  -1.737001   7.440888  -2.809018  -4.698614   \n",
       "496  -3.103700   5.372698  -1.248596   0.087593  -0.029414  -1.633409   \n",
       "497   5.083205  -5.636598  -3.119192  -8.028303   7.971191   2.713695   \n",
       "498  -1.139614  -8.434016  -1.144310  -7.514122  -3.672928  -1.294024   \n",
       "499  -2.065601  -1.242903  -3.652198   1.320391   2.661285  -1.027211   \n",
       "\n",
       "          6         7         8          9    ...       490       491  \\\n",
       "0    0.005601  1.450910  0.406392   1.579916  ...  0.593088  3.850108   \n",
       "1    6.002499  4.607007 -2.434111   2.604113  ...  0.910285  1.025905   \n",
       "2    3.053404  9.695713 -6.101305   3.928019  ...  6.035491  5.882611   \n",
       "3    7.369393  7.878902 -6.396716  -0.043792  ...  4.393280  5.632500   \n",
       "4   -1.699513 -4.664904  5.307878   4.027002  ...  2.649374  6.923194   \n",
       "..        ...       ...       ...        ...  ...       ...       ...   \n",
       "495  1.355193 -0.148798  3.515184   8.046708  ... -0.764420 -4.210900   \n",
       "496 -1.137002  0.502606 -8.135812  -5.761088  ...  2.460984  2.056504   \n",
       "497 -6.155798 -6.842089 -4.205207 -12.150483  ... -2.619311  4.996009   \n",
       "498 -4.383217 -4.969508  2.194474  -4.847502  ... -4.139230 -8.862910   \n",
       "499 -6.002004 -8.532495 -2.822213  -5.130689  ... -2.928217 -2.104897   \n",
       "\n",
       "          492        493        494        495        496        497  \\\n",
       "0   -9.136295   4.292013  -3.399712 -10.672104  -3.103700   5.083205   \n",
       "1   -2.711797   0.154310   0.319285   8.139194   5.372698  -5.636598   \n",
       "2    9.458008  -6.592084   5.767391  -1.737001  -1.248596  -3.119192   \n",
       "3    7.720597   0.812005   2.467780   7.440888   0.087593  -8.028303   \n",
       "4   -7.399309   0.623699   1.256074  -2.809018  -0.029414   7.971191   \n",
       "..        ...        ...        ...        ...        ...        ...   \n",
       "495  3.462097  -6.719095  18.034280  40.463588   7.860093  -8.158003   \n",
       "496  3.404502  -1.032691  -0.650716   7.860093  29.333497  19.186501   \n",
       "497  6.710706  11.294114  -6.044211  -8.158003  19.186501  36.603206   \n",
       "498  6.183287  -2.552405   6.351270   4.955378   0.829183   9.409287   \n",
       "499 -0.786400   1.265808   1.119183   8.036091  -3.456904   3.965700   \n",
       "\n",
       "           498        499  \n",
       "0    -1.139614  -2.065601  \n",
       "1    -8.434016  -1.242903  \n",
       "2    -1.144310  -3.652198  \n",
       "3    -7.514122   1.320391  \n",
       "4    -3.672928   2.661285  \n",
       "..         ...        ...  \n",
       "495   4.955378   8.036091  \n",
       "496   0.829183  -3.456904  \n",
       "497   9.409287   3.965700  \n",
       "498  33.992669  24.624281  \n",
       "499  24.624281  51.905994  \n",
       "\n",
       "[500 rows x 500 columns]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K_c"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6f345ff9",
   "metadata": {},
   "source": [
    "## Manual EigenValues and EigenVector"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "bc12e1c7",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
       "      <th>4</th>\n",
       "      <th>5</th>\n",
       "      <th>6</th>\n",
       "      <th>7</th>\n",
       "      <th>8</th>\n",
       "      <th>9</th>\n",
       "      <th>...</th>\n",
       "      <th>490</th>\n",
       "      <th>491</th>\n",
       "      <th>492</th>\n",
       "      <th>493</th>\n",
       "      <th>494</th>\n",
       "      <th>495</th>\n",
       "      <th>496</th>\n",
       "      <th>497</th>\n",
       "      <th>498</th>\n",
       "      <th>499</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>11.254804</td>\n",
       "      <td>-1.006399</td>\n",
       "      <td>0.067607</td>\n",
       "      <td>-2.911104</td>\n",
       "      <td>7.180790</td>\n",
       "      <td>2.801094</td>\n",
       "      <td>0.005601</td>\n",
       "      <td>1.450910</td>\n",
       "      <td>0.406392</td>\n",
       "      <td>1.579916</td>\n",
       "      <td>...</td>\n",
       "      <td>0.593088</td>\n",
       "      <td>3.850108</td>\n",
       "      <td>-9.136295</td>\n",
       "      <td>4.292013</td>\n",
       "      <td>-3.399712</td>\n",
       "      <td>-10.672104</td>\n",
       "      <td>-3.103700</td>\n",
       "      <td>5.083205</td>\n",
       "      <td>-1.139614</td>\n",
       "      <td>-2.065601</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-1.006399</td>\n",
       "      <td>15.648399</td>\n",
       "      <td>-1.643795</td>\n",
       "      <td>5.655894</td>\n",
       "      <td>2.590487</td>\n",
       "      <td>-0.440508</td>\n",
       "      <td>6.002499</td>\n",
       "      <td>4.607007</td>\n",
       "      <td>-2.434111</td>\n",
       "      <td>2.604113</td>\n",
       "      <td>...</td>\n",
       "      <td>0.910285</td>\n",
       "      <td>1.025905</td>\n",
       "      <td>-2.711797</td>\n",
       "      <td>0.154310</td>\n",
       "      <td>0.319285</td>\n",
       "      <td>8.139194</td>\n",
       "      <td>5.372698</td>\n",
       "      <td>-5.636598</td>\n",
       "      <td>-8.434016</td>\n",
       "      <td>-1.242903</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.067607</td>\n",
       "      <td>-1.643795</td>\n",
       "      <td>26.753411</td>\n",
       "      <td>17.878799</td>\n",
       "      <td>-0.314207</td>\n",
       "      <td>5.125297</td>\n",
       "      <td>3.053404</td>\n",
       "      <td>9.695713</td>\n",
       "      <td>-6.101305</td>\n",
       "      <td>3.928019</td>\n",
       "      <td>...</td>\n",
       "      <td>6.035491</td>\n",
       "      <td>5.882611</td>\n",
       "      <td>9.458008</td>\n",
       "      <td>-6.592084</td>\n",
       "      <td>5.767391</td>\n",
       "      <td>-1.737001</td>\n",
       "      <td>-1.248596</td>\n",
       "      <td>-3.119192</td>\n",
       "      <td>-1.144310</td>\n",
       "      <td>-3.652198</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-2.911104</td>\n",
       "      <td>5.655894</td>\n",
       "      <td>17.878799</td>\n",
       "      <td>28.075988</td>\n",
       "      <td>-1.217818</td>\n",
       "      <td>1.028286</td>\n",
       "      <td>7.369393</td>\n",
       "      <td>7.878902</td>\n",
       "      <td>-6.396716</td>\n",
       "      <td>-0.043792</td>\n",
       "      <td>...</td>\n",
       "      <td>4.393280</td>\n",
       "      <td>5.632500</td>\n",
       "      <td>7.720597</td>\n",
       "      <td>0.812005</td>\n",
       "      <td>2.467780</td>\n",
       "      <td>7.440888</td>\n",
       "      <td>0.087593</td>\n",
       "      <td>-8.028303</td>\n",
       "      <td>-7.514122</td>\n",
       "      <td>1.320391</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>7.180790</td>\n",
       "      <td>2.590487</td>\n",
       "      <td>-0.314207</td>\n",
       "      <td>-1.217818</td>\n",
       "      <td>40.462976</td>\n",
       "      <td>34.783680</td>\n",
       "      <td>-1.699513</td>\n",
       "      <td>-4.664904</td>\n",
       "      <td>5.307878</td>\n",
       "      <td>4.027002</td>\n",
       "      <td>...</td>\n",
       "      <td>2.649374</td>\n",
       "      <td>6.923194</td>\n",
       "      <td>-7.399309</td>\n",
       "      <td>0.623699</td>\n",
       "      <td>1.256074</td>\n",
       "      <td>-2.809018</td>\n",
       "      <td>-0.029414</td>\n",
       "      <td>7.971191</td>\n",
       "      <td>-3.672928</td>\n",
       "      <td>2.661285</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>495</th>\n",
       "      <td>-10.672104</td>\n",
       "      <td>8.139194</td>\n",
       "      <td>-1.737001</td>\n",
       "      <td>7.440888</td>\n",
       "      <td>-2.809018</td>\n",
       "      <td>-4.698614</td>\n",
       "      <td>1.355193</td>\n",
       "      <td>-0.148798</td>\n",
       "      <td>3.515184</td>\n",
       "      <td>8.046708</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.764420</td>\n",
       "      <td>-4.210900</td>\n",
       "      <td>3.462097</td>\n",
       "      <td>-6.719095</td>\n",
       "      <td>18.034280</td>\n",
       "      <td>40.463588</td>\n",
       "      <td>7.860093</td>\n",
       "      <td>-8.158003</td>\n",
       "      <td>4.955378</td>\n",
       "      <td>8.036091</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>496</th>\n",
       "      <td>-3.103700</td>\n",
       "      <td>5.372698</td>\n",
       "      <td>-1.248596</td>\n",
       "      <td>0.087593</td>\n",
       "      <td>-0.029414</td>\n",
       "      <td>-1.633409</td>\n",
       "      <td>-1.137002</td>\n",
       "      <td>0.502606</td>\n",
       "      <td>-8.135812</td>\n",
       "      <td>-5.761088</td>\n",
       "      <td>...</td>\n",
       "      <td>2.460984</td>\n",
       "      <td>2.056504</td>\n",
       "      <td>3.404502</td>\n",
       "      <td>-1.032691</td>\n",
       "      <td>-0.650716</td>\n",
       "      <td>7.860093</td>\n",
       "      <td>29.333497</td>\n",
       "      <td>19.186501</td>\n",
       "      <td>0.829183</td>\n",
       "      <td>-3.456904</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>497</th>\n",
       "      <td>5.083205</td>\n",
       "      <td>-5.636598</td>\n",
       "      <td>-3.119192</td>\n",
       "      <td>-8.028303</td>\n",
       "      <td>7.971191</td>\n",
       "      <td>2.713695</td>\n",
       "      <td>-6.155798</td>\n",
       "      <td>-6.842089</td>\n",
       "      <td>-4.205207</td>\n",
       "      <td>-12.150483</td>\n",
       "      <td>...</td>\n",
       "      <td>-2.619311</td>\n",
       "      <td>4.996009</td>\n",
       "      <td>6.710706</td>\n",
       "      <td>11.294114</td>\n",
       "      <td>-6.044211</td>\n",
       "      <td>-8.158003</td>\n",
       "      <td>19.186501</td>\n",
       "      <td>36.603206</td>\n",
       "      <td>9.409287</td>\n",
       "      <td>3.965700</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>498</th>\n",
       "      <td>-1.139614</td>\n",
       "      <td>-8.434016</td>\n",
       "      <td>-1.144310</td>\n",
       "      <td>-7.514122</td>\n",
       "      <td>-3.672928</td>\n",
       "      <td>-1.294024</td>\n",
       "      <td>-4.383217</td>\n",
       "      <td>-4.969508</td>\n",
       "      <td>2.194474</td>\n",
       "      <td>-4.847502</td>\n",
       "      <td>...</td>\n",
       "      <td>-4.139230</td>\n",
       "      <td>-8.862910</td>\n",
       "      <td>6.183287</td>\n",
       "      <td>-2.552405</td>\n",
       "      <td>6.351270</td>\n",
       "      <td>4.955378</td>\n",
       "      <td>0.829183</td>\n",
       "      <td>9.409287</td>\n",
       "      <td>33.992669</td>\n",
       "      <td>24.624281</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>499</th>\n",
       "      <td>-2.065601</td>\n",
       "      <td>-1.242903</td>\n",
       "      <td>-3.652198</td>\n",
       "      <td>1.320391</td>\n",
       "      <td>2.661285</td>\n",
       "      <td>-1.027211</td>\n",
       "      <td>-6.002004</td>\n",
       "      <td>-8.532495</td>\n",
       "      <td>-2.822213</td>\n",
       "      <td>-5.130689</td>\n",
       "      <td>...</td>\n",
       "      <td>-2.928217</td>\n",
       "      <td>-2.104897</td>\n",
       "      <td>-0.786400</td>\n",
       "      <td>1.265808</td>\n",
       "      <td>1.119183</td>\n",
       "      <td>8.036091</td>\n",
       "      <td>-3.456904</td>\n",
       "      <td>3.965700</td>\n",
       "      <td>24.624281</td>\n",
       "      <td>51.905994</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>500 rows × 500 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "           0          1          2          3          4          5    \\\n",
       "0    11.254804  -1.006399   0.067607  -2.911104   7.180790   2.801094   \n",
       "1    -1.006399  15.648399  -1.643795   5.655894   2.590487  -0.440508   \n",
       "2     0.067607  -1.643795  26.753411  17.878799  -0.314207   5.125297   \n",
       "3    -2.911104   5.655894  17.878799  28.075988  -1.217818   1.028286   \n",
       "4     7.180790   2.590487  -0.314207  -1.217818  40.462976  34.783680   \n",
       "..         ...        ...        ...        ...        ...        ...   \n",
       "495 -10.672104   8.139194  -1.737001   7.440888  -2.809018  -4.698614   \n",
       "496  -3.103700   5.372698  -1.248596   0.087593  -0.029414  -1.633409   \n",
       "497   5.083205  -5.636598  -3.119192  -8.028303   7.971191   2.713695   \n",
       "498  -1.139614  -8.434016  -1.144310  -7.514122  -3.672928  -1.294024   \n",
       "499  -2.065601  -1.242903  -3.652198   1.320391   2.661285  -1.027211   \n",
       "\n",
       "          6         7         8          9    ...       490       491  \\\n",
       "0    0.005601  1.450910  0.406392   1.579916  ...  0.593088  3.850108   \n",
       "1    6.002499  4.607007 -2.434111   2.604113  ...  0.910285  1.025905   \n",
       "2    3.053404  9.695713 -6.101305   3.928019  ...  6.035491  5.882611   \n",
       "3    7.369393  7.878902 -6.396716  -0.043792  ...  4.393280  5.632500   \n",
       "4   -1.699513 -4.664904  5.307878   4.027002  ...  2.649374  6.923194   \n",
       "..        ...       ...       ...        ...  ...       ...       ...   \n",
       "495  1.355193 -0.148798  3.515184   8.046708  ... -0.764420 -4.210900   \n",
       "496 -1.137002  0.502606 -8.135812  -5.761088  ...  2.460984  2.056504   \n",
       "497 -6.155798 -6.842089 -4.205207 -12.150483  ... -2.619311  4.996009   \n",
       "498 -4.383217 -4.969508  2.194474  -4.847502  ... -4.139230 -8.862910   \n",
       "499 -6.002004 -8.532495 -2.822213  -5.130689  ... -2.928217 -2.104897   \n",
       "\n",
       "          492        493        494        495        496        497  \\\n",
       "0   -9.136295   4.292013  -3.399712 -10.672104  -3.103700   5.083205   \n",
       "1   -2.711797   0.154310   0.319285   8.139194   5.372698  -5.636598   \n",
       "2    9.458008  -6.592084   5.767391  -1.737001  -1.248596  -3.119192   \n",
       "3    7.720597   0.812005   2.467780   7.440888   0.087593  -8.028303   \n",
       "4   -7.399309   0.623699   1.256074  -2.809018  -0.029414   7.971191   \n",
       "..        ...        ...        ...        ...        ...        ...   \n",
       "495  3.462097  -6.719095  18.034280  40.463588   7.860093  -8.158003   \n",
       "496  3.404502  -1.032691  -0.650716   7.860093  29.333497  19.186501   \n",
       "497  6.710706  11.294114  -6.044211  -8.158003  19.186501  36.603206   \n",
       "498  6.183287  -2.552405   6.351270   4.955378   0.829183   9.409287   \n",
       "499 -0.786400   1.265808   1.119183   8.036091  -3.456904   3.965700   \n",
       "\n",
       "           498        499  \n",
       "0    -1.139614  -2.065601  \n",
       "1    -8.434016  -1.242903  \n",
       "2    -1.144310  -3.652198  \n",
       "3    -7.514122   1.320391  \n",
       "4    -3.672928   2.661285  \n",
       "..         ...        ...  \n",
       "495   4.955378   8.036091  \n",
       "496   0.829183  -3.456904  \n",
       "497   9.409287   3.965700  \n",
       "498  33.992669  24.624281  \n",
       "499  24.624281  51.905994  \n",
       "\n",
       "[500 rows x 500 columns]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K_c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "490c7555",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
       "      <th>4</th>\n",
       "      <th>5</th>\n",
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       "      <th>8</th>\n",
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       "      <th>490</th>\n",
       "      <th>491</th>\n",
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       "      <th>493</th>\n",
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       "      <td>0.000135</td>\n",
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       "      <td>0.014362</td>\n",
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       "      <td>0.002902</td>\n",
       "      <td>0.000813</td>\n",
       "      <td>0.003160</td>\n",
       "      <td>...</td>\n",
       "      <td>0.001186</td>\n",
       "      <td>0.007700</td>\n",
       "      <td>-0.018273</td>\n",
       "      <td>0.008584</td>\n",
       "      <td>-0.006799</td>\n",
       "      <td>-0.021344</td>\n",
       "      <td>-0.006207</td>\n",
       "      <td>0.010166</td>\n",
       "      <td>-0.002279</td>\n",
       "      <td>-0.004131</td>\n",
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       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-0.002013</td>\n",
       "      <td>0.031297</td>\n",
       "      <td>-0.003288</td>\n",
       "      <td>0.011312</td>\n",
       "      <td>0.005181</td>\n",
       "      <td>-0.000881</td>\n",
       "      <td>0.012005</td>\n",
       "      <td>0.009214</td>\n",
       "      <td>-0.004868</td>\n",
       "      <td>0.005208</td>\n",
       "      <td>...</td>\n",
       "      <td>0.001821</td>\n",
       "      <td>0.002052</td>\n",
       "      <td>-0.005424</td>\n",
       "      <td>0.000309</td>\n",
       "      <td>0.000639</td>\n",
       "      <td>0.016278</td>\n",
       "      <td>0.010745</td>\n",
       "      <td>-0.011273</td>\n",
       "      <td>-0.016868</td>\n",
       "      <td>-0.002486</td>\n",
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       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.000135</td>\n",
       "      <td>-0.003288</td>\n",
       "      <td>0.053507</td>\n",
       "      <td>0.035758</td>\n",
       "      <td>-0.000628</td>\n",
       "      <td>0.010251</td>\n",
       "      <td>0.006107</td>\n",
       "      <td>0.019391</td>\n",
       "      <td>-0.012203</td>\n",
       "      <td>0.007856</td>\n",
       "      <td>...</td>\n",
       "      <td>0.012071</td>\n",
       "      <td>0.011765</td>\n",
       "      <td>0.018916</td>\n",
       "      <td>-0.013184</td>\n",
       "      <td>0.011535</td>\n",
       "      <td>-0.003474</td>\n",
       "      <td>-0.002497</td>\n",
       "      <td>-0.006238</td>\n",
       "      <td>-0.002289</td>\n",
       "      <td>-0.007304</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-0.005822</td>\n",
       "      <td>0.011312</td>\n",
       "      <td>0.035758</td>\n",
       "      <td>0.056152</td>\n",
       "      <td>-0.002436</td>\n",
       "      <td>0.002057</td>\n",
       "      <td>0.014739</td>\n",
       "      <td>0.015758</td>\n",
       "      <td>-0.012793</td>\n",
       "      <td>-0.000088</td>\n",
       "      <td>...</td>\n",
       "      <td>0.008787</td>\n",
       "      <td>0.011265</td>\n",
       "      <td>0.015441</td>\n",
       "      <td>0.001624</td>\n",
       "      <td>0.004936</td>\n",
       "      <td>0.014882</td>\n",
       "      <td>0.000175</td>\n",
       "      <td>-0.016057</td>\n",
       "      <td>-0.015028</td>\n",
       "      <td>0.002641</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.014362</td>\n",
       "      <td>0.005181</td>\n",
       "      <td>-0.000628</td>\n",
       "      <td>-0.002436</td>\n",
       "      <td>0.080926</td>\n",
       "      <td>0.069567</td>\n",
       "      <td>-0.003399</td>\n",
       "      <td>-0.009330</td>\n",
       "      <td>0.010616</td>\n",
       "      <td>0.008054</td>\n",
       "      <td>...</td>\n",
       "      <td>0.005299</td>\n",
       "      <td>0.013846</td>\n",
       "      <td>-0.014799</td>\n",
       "      <td>0.001247</td>\n",
       "      <td>0.002512</td>\n",
       "      <td>-0.005618</td>\n",
       "      <td>-0.000059</td>\n",
       "      <td>0.015942</td>\n",
       "      <td>-0.007346</td>\n",
       "      <td>0.005323</td>\n",
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       "      <td>...</td>\n",
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       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
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       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
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       "      <th>495</th>\n",
       "      <td>-0.021344</td>\n",
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       "      <td>-0.003474</td>\n",
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       "      <td>-0.005618</td>\n",
       "      <td>-0.009397</td>\n",
       "      <td>0.002710</td>\n",
       "      <td>-0.000298</td>\n",
       "      <td>0.007030</td>\n",
       "      <td>0.016093</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.001529</td>\n",
       "      <td>-0.008422</td>\n",
       "      <td>0.006924</td>\n",
       "      <td>-0.013438</td>\n",
       "      <td>0.036069</td>\n",
       "      <td>0.080927</td>\n",
       "      <td>0.015720</td>\n",
       "      <td>-0.016316</td>\n",
       "      <td>0.009911</td>\n",
       "      <td>0.016072</td>\n",
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       "    <tr>\n",
       "      <th>496</th>\n",
       "      <td>-0.006207</td>\n",
       "      <td>0.010745</td>\n",
       "      <td>-0.002497</td>\n",
       "      <td>0.000175</td>\n",
       "      <td>-0.000059</td>\n",
       "      <td>-0.003267</td>\n",
       "      <td>-0.002274</td>\n",
       "      <td>0.001005</td>\n",
       "      <td>-0.016272</td>\n",
       "      <td>-0.011522</td>\n",
       "      <td>...</td>\n",
       "      <td>0.004922</td>\n",
       "      <td>0.004113</td>\n",
       "      <td>0.006809</td>\n",
       "      <td>-0.002065</td>\n",
       "      <td>-0.001301</td>\n",
       "      <td>0.015720</td>\n",
       "      <td>0.058667</td>\n",
       "      <td>0.038373</td>\n",
       "      <td>0.001658</td>\n",
       "      <td>-0.006914</td>\n",
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       "    <tr>\n",
       "      <th>497</th>\n",
       "      <td>0.010166</td>\n",
       "      <td>-0.011273</td>\n",
       "      <td>-0.006238</td>\n",
       "      <td>-0.016057</td>\n",
       "      <td>0.015942</td>\n",
       "      <td>0.005427</td>\n",
       "      <td>-0.012312</td>\n",
       "      <td>-0.013684</td>\n",
       "      <td>-0.008410</td>\n",
       "      <td>-0.024301</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.005239</td>\n",
       "      <td>0.009992</td>\n",
       "      <td>0.013421</td>\n",
       "      <td>0.022588</td>\n",
       "      <td>-0.012088</td>\n",
       "      <td>-0.016316</td>\n",
       "      <td>0.038373</td>\n",
       "      <td>0.073206</td>\n",
       "      <td>0.018819</td>\n",
       "      <td>0.007931</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>498</th>\n",
       "      <td>-0.002279</td>\n",
       "      <td>-0.016868</td>\n",
       "      <td>-0.002289</td>\n",
       "      <td>-0.015028</td>\n",
       "      <td>-0.007346</td>\n",
       "      <td>-0.002588</td>\n",
       "      <td>-0.008766</td>\n",
       "      <td>-0.009939</td>\n",
       "      <td>0.004389</td>\n",
       "      <td>-0.009695</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.008278</td>\n",
       "      <td>-0.017726</td>\n",
       "      <td>0.012367</td>\n",
       "      <td>-0.005105</td>\n",
       "      <td>0.012703</td>\n",
       "      <td>0.009911</td>\n",
       "      <td>0.001658</td>\n",
       "      <td>0.018819</td>\n",
       "      <td>0.067985</td>\n",
       "      <td>0.049249</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>499</th>\n",
       "      <td>-0.004131</td>\n",
       "      <td>-0.002486</td>\n",
       "      <td>-0.007304</td>\n",
       "      <td>0.002641</td>\n",
       "      <td>0.005323</td>\n",
       "      <td>-0.002054</td>\n",
       "      <td>-0.012004</td>\n",
       "      <td>-0.017065</td>\n",
       "      <td>-0.005644</td>\n",
       "      <td>-0.010261</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.005856</td>\n",
       "      <td>-0.004210</td>\n",
       "      <td>-0.001573</td>\n",
       "      <td>0.002532</td>\n",
       "      <td>0.002238</td>\n",
       "      <td>0.016072</td>\n",
       "      <td>-0.006914</td>\n",
       "      <td>0.007931</td>\n",
       "      <td>0.049249</td>\n",
       "      <td>0.103812</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>500 rows × 500 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "          0         1         2         3         4         5         6    \\\n",
       "0    0.022510 -0.002013  0.000135 -0.005822  0.014362  0.005602  0.000011   \n",
       "1   -0.002013  0.031297 -0.003288  0.011312  0.005181 -0.000881  0.012005   \n",
       "2    0.000135 -0.003288  0.053507  0.035758 -0.000628  0.010251  0.006107   \n",
       "3   -0.005822  0.011312  0.035758  0.056152 -0.002436  0.002057  0.014739   \n",
       "4    0.014362  0.005181 -0.000628 -0.002436  0.080926  0.069567 -0.003399   \n",
       "..        ...       ...       ...       ...       ...       ...       ...   \n",
       "495 -0.021344  0.016278 -0.003474  0.014882 -0.005618 -0.009397  0.002710   \n",
       "496 -0.006207  0.010745 -0.002497  0.000175 -0.000059 -0.003267 -0.002274   \n",
       "497  0.010166 -0.011273 -0.006238 -0.016057  0.015942  0.005427 -0.012312   \n",
       "498 -0.002279 -0.016868 -0.002289 -0.015028 -0.007346 -0.002588 -0.008766   \n",
       "499 -0.004131 -0.002486 -0.007304  0.002641  0.005323 -0.002054 -0.012004   \n",
       "\n",
       "          7         8         9    ...       490       491       492  \\\n",
       "0    0.002902  0.000813  0.003160  ...  0.001186  0.007700 -0.018273   \n",
       "1    0.009214 -0.004868  0.005208  ...  0.001821  0.002052 -0.005424   \n",
       "2    0.019391 -0.012203  0.007856  ...  0.012071  0.011765  0.018916   \n",
       "3    0.015758 -0.012793 -0.000088  ...  0.008787  0.011265  0.015441   \n",
       "4   -0.009330  0.010616  0.008054  ...  0.005299  0.013846 -0.014799   \n",
       "..        ...       ...       ...  ...       ...       ...       ...   \n",
       "495 -0.000298  0.007030  0.016093  ... -0.001529 -0.008422  0.006924   \n",
       "496  0.001005 -0.016272 -0.011522  ...  0.004922  0.004113  0.006809   \n",
       "497 -0.013684 -0.008410 -0.024301  ... -0.005239  0.009992  0.013421   \n",
       "498 -0.009939  0.004389 -0.009695  ... -0.008278 -0.017726  0.012367   \n",
       "499 -0.017065 -0.005644 -0.010261  ... -0.005856 -0.004210 -0.001573   \n",
       "\n",
       "          493       494       495       496       497       498       499  \n",
       "0    0.008584 -0.006799 -0.021344 -0.006207  0.010166 -0.002279 -0.004131  \n",
       "1    0.000309  0.000639  0.016278  0.010745 -0.011273 -0.016868 -0.002486  \n",
       "2   -0.013184  0.011535 -0.003474 -0.002497 -0.006238 -0.002289 -0.007304  \n",
       "3    0.001624  0.004936  0.014882  0.000175 -0.016057 -0.015028  0.002641  \n",
       "4    0.001247  0.002512 -0.005618 -0.000059  0.015942 -0.007346  0.005323  \n",
       "..        ...       ...       ...       ...       ...       ...       ...  \n",
       "495 -0.013438  0.036069  0.080927  0.015720 -0.016316  0.009911  0.016072  \n",
       "496 -0.002065 -0.001301  0.015720  0.058667  0.038373  0.001658 -0.006914  \n",
       "497  0.022588 -0.012088 -0.016316  0.038373  0.073206  0.018819  0.007931  \n",
       "498 -0.005105  0.012703  0.009911  0.001658  0.018819  0.067985  0.049249  \n",
       "499  0.002532  0.002238  0.016072 -0.006914  0.007931  0.049249  0.103812  \n",
       "\n",
       "[500 rows x 500 columns]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K_c/len(df_scl)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "d8f1146c",
   "metadata": {},
   "outputs": [],
   "source": [
    "u, s, vh = np.linalg.svd(K_c/len(df_scl))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "b11ad4db",
   "metadata": {},
   "outputs": [],
   "source": [
    "u = pd.DataFrame(u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "43efba06",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
       "      <th>4</th>\n",
       "      <th>5</th>\n",
       "      <th>6</th>\n",
       "      <th>7</th>\n",
       "      <th>8</th>\n",
       "      <th>9</th>\n",
       "      <th>...</th>\n",
       "      <th>490</th>\n",
       "      <th>491</th>\n",
       "      <th>492</th>\n",
       "      <th>493</th>\n",
       "      <th>494</th>\n",
       "      <th>495</th>\n",
       "      <th>496</th>\n",
       "      <th>497</th>\n",
       "      <th>498</th>\n",
       "      <th>499</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-0.006387</td>\n",
       "      <td>0.023537</td>\n",
       "      <td>-0.004673</td>\n",
       "      <td>0.027282</td>\n",
       "      <td>-0.048683</td>\n",
       "      <td>0.001987</td>\n",
       "      <td>-0.036125</td>\n",
       "      <td>0.017959</td>\n",
       "      <td>-0.008994</td>\n",
       "      <td>0.032167</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.050937</td>\n",
       "      <td>0.041807</td>\n",
       "      <td>-0.043418</td>\n",
       "      <td>0.005216</td>\n",
       "      <td>-0.037156</td>\n",
       "      <td>0.011797</td>\n",
       "      <td>-0.027347</td>\n",
       "      <td>0.005959</td>\n",
       "      <td>-0.039187</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-0.010837</td>\n",
       "      <td>0.005218</td>\n",
       "      <td>0.014919</td>\n",
       "      <td>-0.048530</td>\n",
       "      <td>-0.016666</td>\n",
       "      <td>0.002775</td>\n",
       "      <td>0.009661</td>\n",
       "      <td>-0.019979</td>\n",
       "      <td>0.034658</td>\n",
       "      <td>-0.048371</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.052567</td>\n",
       "      <td>-0.003085</td>\n",
       "      <td>-0.015609</td>\n",
       "      <td>-0.082312</td>\n",
       "      <td>0.042567</td>\n",
       "      <td>0.070397</td>\n",
       "      <td>-0.067255</td>\n",
       "      <td>-0.004390</td>\n",
       "      <td>0.052487</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-0.009057</td>\n",
       "      <td>0.006176</td>\n",
       "      <td>0.027448</td>\n",
       "      <td>0.044896</td>\n",
       "      <td>-0.030452</td>\n",
       "      <td>-0.072360</td>\n",
       "      <td>0.011022</td>\n",
       "      <td>-0.010565</td>\n",
       "      <td>0.032952</td>\n",
       "      <td>-0.039450</td>\n",
       "      <td>...</td>\n",
       "      <td>0.004636</td>\n",
       "      <td>-0.046356</td>\n",
       "      <td>-0.023842</td>\n",
       "      <td>-0.024329</td>\n",
       "      <td>0.040072</td>\n",
       "      <td>0.008014</td>\n",
       "      <td>0.006137</td>\n",
       "      <td>-0.099489</td>\n",
       "      <td>0.025765</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-0.010602</td>\n",
       "      <td>0.003314</td>\n",
       "      <td>0.021919</td>\n",
       "      <td>0.007954</td>\n",
       "      <td>-0.000284</td>\n",
       "      <td>-0.025181</td>\n",
       "      <td>0.000550</td>\n",
       "      <td>0.002075</td>\n",
       "      <td>0.055837</td>\n",
       "      <td>-0.063263</td>\n",
       "      <td>...</td>\n",
       "      <td>0.001097</td>\n",
       "      <td>0.022137</td>\n",
       "      <td>-0.016339</td>\n",
       "      <td>0.050687</td>\n",
       "      <td>-0.083193</td>\n",
       "      <td>0.041899</td>\n",
       "      <td>-0.013406</td>\n",
       "      <td>-0.034253</td>\n",
       "      <td>0.014198</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-0.017506</td>\n",
       "      <td>0.042395</td>\n",
       "      <td>0.023584</td>\n",
       "      <td>0.022463</td>\n",
       "      <td>-0.041785</td>\n",
       "      <td>0.051631</td>\n",
       "      <td>-0.039164</td>\n",
       "      <td>0.036588</td>\n",
       "      <td>-0.051554</td>\n",
       "      <td>-0.017074</td>\n",
       "      <td>...</td>\n",
       "      <td>0.004793</td>\n",
       "      <td>-0.101694</td>\n",
       "      <td>-0.037562</td>\n",
       "      <td>0.037930</td>\n",
       "      <td>-0.098179</td>\n",
       "      <td>0.006870</td>\n",
       "      <td>0.038308</td>\n",
       "      <td>0.023583</td>\n",
       "      <td>0.044797</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>495</th>\n",
       "      <td>-0.015570</td>\n",
       "      <td>-0.035488</td>\n",
       "      <td>0.021553</td>\n",
       "      <td>-0.036509</td>\n",
       "      <td>0.071049</td>\n",
       "      <td>0.038697</td>\n",
       "      <td>0.063405</td>\n",
       "      <td>0.004241</td>\n",
       "      <td>0.037852</td>\n",
       "      <td>-0.082169</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.037931</td>\n",
       "      <td>0.004436</td>\n",
       "      <td>-0.019700</td>\n",
       "      <td>-0.021188</td>\n",
       "      <td>0.028748</td>\n",
       "      <td>0.064265</td>\n",
       "      <td>-0.015115</td>\n",
       "      <td>-0.079277</td>\n",
       "      <td>0.016957</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>496</th>\n",
       "      <td>-0.003109</td>\n",
       "      <td>-0.035706</td>\n",
       "      <td>0.000365</td>\n",
       "      <td>-0.074290</td>\n",
       "      <td>-0.008611</td>\n",
       "      <td>-0.058478</td>\n",
       "      <td>-0.036679</td>\n",
       "      <td>-0.032770</td>\n",
       "      <td>0.021545</td>\n",
       "      <td>-0.004079</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.037572</td>\n",
       "      <td>-0.023772</td>\n",
       "      <td>0.056510</td>\n",
       "      <td>-0.007092</td>\n",
       "      <td>-0.069547</td>\n",
       "      <td>-0.085795</td>\n",
       "      <td>-0.018858</td>\n",
       "      <td>-0.029559</td>\n",
       "      <td>0.017190</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>497</th>\n",
       "      <td>0.011545</td>\n",
       "      <td>-0.017246</td>\n",
       "      <td>-0.037298</td>\n",
       "      <td>-0.015249</td>\n",
       "      <td>-0.027140</td>\n",
       "      <td>-0.048940</td>\n",
       "      <td>-0.110584</td>\n",
       "      <td>0.014648</td>\n",
       "      <td>-0.031744</td>\n",
       "      <td>0.004925</td>\n",
       "      <td>...</td>\n",
       "      <td>0.026626</td>\n",
       "      <td>-0.015974</td>\n",
       "      <td>-0.028354</td>\n",
       "      <td>0.054069</td>\n",
       "      <td>0.061694</td>\n",
       "      <td>-0.050347</td>\n",
       "      <td>-0.031000</td>\n",
       "      <td>-0.001908</td>\n",
       "      <td>-0.065878</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>498</th>\n",
       "      <td>-0.002565</td>\n",
       "      <td>-0.008398</td>\n",
       "      <td>-0.042501</td>\n",
       "      <td>0.032443</td>\n",
       "      <td>0.023240</td>\n",
       "      <td>-0.018927</td>\n",
       "      <td>-0.011876</td>\n",
       "      <td>-0.003614</td>\n",
       "      <td>-0.064240</td>\n",
       "      <td>0.003310</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.067470</td>\n",
       "      <td>0.027025</td>\n",
       "      <td>0.043941</td>\n",
       "      <td>-0.022657</td>\n",
       "      <td>-0.044801</td>\n",
       "      <td>-0.040020</td>\n",
       "      <td>0.013491</td>\n",
       "      <td>-0.037706</td>\n",
       "      <td>-0.037748</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>499</th>\n",
       "      <td>-0.011286</td>\n",
       "      <td>-0.008489</td>\n",
       "      <td>-0.023948</td>\n",
       "      <td>0.089253</td>\n",
       "      <td>0.056195</td>\n",
       "      <td>0.025926</td>\n",
       "      <td>-0.058476</td>\n",
       "      <td>0.014983</td>\n",
       "      <td>-0.013383</td>\n",
       "      <td>-0.027448</td>\n",
       "      <td>...</td>\n",
       "      <td>0.079619</td>\n",
       "      <td>-0.068023</td>\n",
       "      <td>-0.073420</td>\n",
       "      <td>-0.032935</td>\n",
       "      <td>0.025839</td>\n",
       "      <td>0.022778</td>\n",
       "      <td>-0.001411</td>\n",
       "      <td>0.057886</td>\n",
       "      <td>-0.003453</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>500 rows × 500 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "          0         1         2         3         4         5         6    \\\n",
       "0   -0.006387  0.023537 -0.004673  0.027282 -0.048683  0.001987 -0.036125   \n",
       "1   -0.010837  0.005218  0.014919 -0.048530 -0.016666  0.002775  0.009661   \n",
       "2   -0.009057  0.006176  0.027448  0.044896 -0.030452 -0.072360  0.011022   \n",
       "3   -0.010602  0.003314  0.021919  0.007954 -0.000284 -0.025181  0.000550   \n",
       "4   -0.017506  0.042395  0.023584  0.022463 -0.041785  0.051631 -0.039164   \n",
       "..        ...       ...       ...       ...       ...       ...       ...   \n",
       "495 -0.015570 -0.035488  0.021553 -0.036509  0.071049  0.038697  0.063405   \n",
       "496 -0.003109 -0.035706  0.000365 -0.074290 -0.008611 -0.058478 -0.036679   \n",
       "497  0.011545 -0.017246 -0.037298 -0.015249 -0.027140 -0.048940 -0.110584   \n",
       "498 -0.002565 -0.008398 -0.042501  0.032443  0.023240 -0.018927 -0.011876   \n",
       "499 -0.011286 -0.008489 -0.023948  0.089253  0.056195  0.025926 -0.058476   \n",
       "\n",
       "          7         8         9    ...       490       491       492  \\\n",
       "0    0.017959 -0.008994  0.032167  ... -0.050937  0.041807 -0.043418   \n",
       "1   -0.019979  0.034658 -0.048371  ... -0.052567 -0.003085 -0.015609   \n",
       "2   -0.010565  0.032952 -0.039450  ...  0.004636 -0.046356 -0.023842   \n",
       "3    0.002075  0.055837 -0.063263  ...  0.001097  0.022137 -0.016339   \n",
       "4    0.036588 -0.051554 -0.017074  ...  0.004793 -0.101694 -0.037562   \n",
       "..        ...       ...       ...  ...       ...       ...       ...   \n",
       "495  0.004241  0.037852 -0.082169  ... -0.037931  0.004436 -0.019700   \n",
       "496 -0.032770  0.021545 -0.004079  ... -0.037572 -0.023772  0.056510   \n",
       "497  0.014648 -0.031744  0.004925  ...  0.026626 -0.015974 -0.028354   \n",
       "498 -0.003614 -0.064240  0.003310  ... -0.067470  0.027025  0.043941   \n",
       "499  0.014983 -0.013383 -0.027448  ...  0.079619 -0.068023 -0.073420   \n",
       "\n",
       "          493       494       495       496       497       498       499  \n",
       "0    0.005216 -0.037156  0.011797 -0.027347  0.005959 -0.039187 -0.044721  \n",
       "1   -0.082312  0.042567  0.070397 -0.067255 -0.004390  0.052487 -0.044721  \n",
       "2   -0.024329  0.040072  0.008014  0.006137 -0.099489  0.025765 -0.044721  \n",
       "3    0.050687 -0.083193  0.041899 -0.013406 -0.034253  0.014198 -0.044721  \n",
       "4    0.037930 -0.098179  0.006870  0.038308  0.023583  0.044797 -0.044721  \n",
       "..        ...       ...       ...       ...       ...       ...       ...  \n",
       "495 -0.021188  0.028748  0.064265 -0.015115 -0.079277  0.016957 -0.044721  \n",
       "496 -0.007092 -0.069547 -0.085795 -0.018858 -0.029559  0.017190 -0.044721  \n",
       "497  0.054069  0.061694 -0.050347 -0.031000 -0.001908 -0.065878 -0.044721  \n",
       "498 -0.022657 -0.044801 -0.040020  0.013491 -0.037706 -0.037748 -0.044721  \n",
       "499 -0.032935  0.025839  0.022778 -0.001411  0.057886 -0.003453 -0.044721  \n",
       "\n",
       "[500 rows x 500 columns]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Unitary arrays (matrix)\n",
    "# where u and the Hermitian \n",
    "# transpose of vh are 2D arrays with orthonormal columns\n",
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "2fa68a1d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(500, 500)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "a3d5f53c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([7.41572790e+00, 4.87716031e+00, 2.91702285e+00, 2.26304325e+00,\n",
       "       2.09174059e+00, 1.96410904e+00, 1.91524406e+00, 1.62854249e+00,\n",
       "       1.52950929e+00, 1.42664381e+00, 1.32478768e+00, 1.27554390e+00,\n",
       "       1.21253926e+00, 1.15699224e+00, 1.14223561e+00, 1.08140859e+00,\n",
       "       1.05831636e+00, 1.04167785e+00, 1.00857664e+00, 9.38895349e-01,\n",
       "       9.11099176e-01, 8.55166277e-01, 8.19187164e-01, 7.94319149e-01,\n",
       "       7.62904166e-01, 7.55881376e-01, 7.15453836e-01, 6.93704506e-01,\n",
       "       6.81613005e-01, 6.28553779e-01, 6.20353194e-01, 5.66408946e-01,\n",
       "       5.48157595e-01, 5.05362928e-01, 4.81325902e-01, 4.50461328e-01,\n",
       "       4.06325635e-01, 3.74823658e-01, 3.45686134e-01, 2.72181785e-01,\n",
       "       2.30627815e-01, 1.44735048e-01, 7.50311976e-02, 4.57077066e-02,\n",
       "       2.07157442e-02, 1.18287831e-02, 6.43691242e-03, 3.91805002e-03,\n",
       "       2.17377496e-03, 8.96794862e-05, 2.45714031e-06, 2.43711394e-06,\n",
       "       2.39359459e-06, 2.37693907e-06, 2.35836338e-06, 2.34921429e-06,\n",
       "       2.33724866e-06, 2.31192126e-06, 2.30232768e-06, 2.29175972e-06,\n",
       "       2.28768589e-06, 2.25931246e-06, 2.25001742e-06, 2.24715395e-06,\n",
       "       2.23014168e-06, 2.22951284e-06, 2.21093561e-06, 2.20820561e-06,\n",
       "       2.18339721e-06, 2.18008319e-06, 2.17625681e-06, 2.16508808e-06,\n",
       "       2.14958523e-06, 2.14923370e-06, 2.12695387e-06, 2.12111467e-06,\n",
       "       2.11449733e-06, 2.11343407e-06, 2.10549075e-06, 2.07697469e-06,\n",
       "       2.07280917e-06, 2.07216415e-06, 2.06774902e-06, 2.06594050e-06,\n",
       "       2.06061598e-06, 2.04956910e-06, 2.04566069e-06, 2.03884480e-06,\n",
       "       2.03842798e-06, 2.02565534e-06, 2.00509118e-06, 2.00317179e-06,\n",
       "       1.98763632e-06, 1.98452155e-06, 1.97803487e-06, 1.97026150e-06,\n",
       "       1.95480320e-06, 1.95122440e-06, 1.94439352e-06, 1.93567914e-06,\n",
       "       1.93236963e-06, 1.93003083e-06, 1.92401038e-06, 1.91673152e-06,\n",
       "       1.90157967e-06, 1.89286958e-06, 1.88965133e-06, 1.87702581e-06,\n",
       "       1.87108149e-06, 1.86622078e-06, 1.85837650e-06, 1.85071269e-06,\n",
       "       1.84598762e-06, 1.83882319e-06, 1.83643024e-06, 1.83057123e-06,\n",
       "       1.83029459e-06, 1.81862365e-06, 1.80927680e-06, 1.80010115e-06,\n",
       "       1.79608888e-06, 1.78684451e-06, 1.77966938e-06, 1.77246896e-06,\n",
       "       1.76787227e-06, 1.76640311e-06, 1.75784663e-06, 1.75390082e-06,\n",
       "       1.75027260e-06, 1.74440783e-06, 1.73767665e-06, 1.73484740e-06,\n",
       "       1.73197554e-06, 1.72949077e-06, 1.72351068e-06, 1.71496376e-06,\n",
       "       1.70320718e-06, 1.68753183e-06, 1.67862282e-06, 1.67801582e-06,\n",
       "       1.67081527e-06, 1.67060939e-06, 1.65720664e-06, 1.65708701e-06,\n",
       "       1.64560456e-06, 1.64476471e-06, 1.64004596e-06, 1.63131880e-06,\n",
       "       1.62894734e-06, 1.62315442e-06, 1.61195611e-06, 1.60366643e-06,\n",
       "       1.59727483e-06, 1.59040509e-06, 1.58651732e-06, 1.58319003e-06,\n",
       "       1.58044106e-06, 1.57544754e-06, 1.57109822e-06, 1.56611012e-06,\n",
       "       1.56607437e-06, 1.55791626e-06, 1.55410906e-06, 1.54659609e-06,\n",
       "       1.54280934e-06, 1.52975471e-06, 1.52799153e-06, 1.52685570e-06,\n",
       "       1.52187915e-06, 1.51961635e-06, 1.51705441e-06, 1.50468559e-06,\n",
       "       1.49616181e-06, 1.48548683e-06, 1.48313252e-06, 1.48126258e-06,\n",
       "       1.47777521e-06, 1.46761264e-06, 1.46504451e-06, 1.46234853e-06,\n",
       "       1.45892850e-06, 1.44808319e-06, 1.44589586e-06, 1.44129570e-06,\n",
       "       1.44011300e-06, 1.43542034e-06, 1.42680821e-06, 1.42635698e-06,\n",
       "       1.42203230e-06, 1.41825049e-06, 1.41051791e-06, 1.40641723e-06,\n",
       "       1.38834066e-06, 1.38310488e-06, 1.37313887e-06, 1.37266076e-06,\n",
       "       1.36596577e-06, 1.35987890e-06, 1.35918149e-06, 1.35416142e-06,\n",
       "       1.34662312e-06, 1.34410881e-06, 1.33589355e-06, 1.33363886e-06,\n",
       "       1.33153972e-06, 1.32739220e-06, 1.31728268e-06, 1.31267121e-06,\n",
       "       1.31252330e-06, 1.29943743e-06, 1.29812489e-06, 1.28847966e-06,\n",
       "       1.28609452e-06, 1.27646105e-06, 1.27552448e-06, 1.27092768e-06,\n",
       "       1.26970916e-06, 1.25799474e-06, 1.25695415e-06, 1.25299550e-06,\n",
       "       1.24214070e-06, 1.24031101e-06, 1.23754506e-06, 1.23298233e-06,\n",
       "       1.22238296e-06, 1.21789301e-06, 1.21416541e-06, 1.21382636e-06,\n",
       "       1.20959369e-06, 1.19567356e-06, 1.19383626e-06, 1.18416358e-06,\n",
       "       1.17908513e-06, 1.17854759e-06, 1.17713341e-06, 1.17221323e-06,\n",
       "       1.16969669e-06, 1.15694348e-06, 1.15246983e-06, 1.15057659e-06,\n",
       "       1.14542583e-06, 1.14416032e-06, 1.14379129e-06, 1.13981272e-06,\n",
       "       1.12506203e-06, 1.12008510e-06, 1.11670997e-06, 1.11106823e-06,\n",
       "       1.10966435e-06, 1.10578644e-06, 1.10072903e-06, 1.09497897e-06,\n",
       "       1.09314930e-06, 1.08830615e-06, 1.08401106e-06, 1.08202703e-06,\n",
       "       1.07568963e-06, 1.06938906e-06, 1.06493675e-06, 1.05703134e-06,\n",
       "       1.05366569e-06, 1.05163487e-06, 1.05027342e-06, 1.03953101e-06,\n",
       "       1.03629895e-06, 1.03392248e-06, 1.02351777e-06, 1.02015570e-06,\n",
       "       1.01615715e-06, 1.00641598e-06, 1.00178061e-06, 9.95992457e-07,\n",
       "       9.92324321e-07, 9.91197789e-07, 9.88179942e-07, 9.76870407e-07,\n",
       "       9.72283325e-07, 9.69993141e-07, 9.65796184e-07, 9.59234422e-07,\n",
       "       9.52664931e-07, 9.49372679e-07, 9.46681202e-07, 9.39255664e-07,\n",
       "       9.34403838e-07, 9.28615166e-07, 9.28356582e-07, 9.19728413e-07,\n",
       "       9.15388996e-07, 9.12558741e-07, 9.09936394e-07, 9.05551680e-07,\n",
       "       8.98795677e-07, 8.93798386e-07, 8.91901087e-07, 8.89882207e-07,\n",
       "       8.82488305e-07, 8.81378125e-07, 8.77814867e-07, 8.77465824e-07,\n",
       "       8.69395142e-07, 8.66280303e-07, 8.56732602e-07, 8.54413949e-07,\n",
       "       8.46602894e-07, 8.45925367e-07, 8.39612851e-07, 8.35541351e-07,\n",
       "       8.32748859e-07, 8.24637929e-07, 8.23678709e-07, 8.16693576e-07,\n",
       "       8.15824376e-07, 8.11936740e-07, 7.98430834e-07, 7.95950672e-07,\n",
       "       7.95112084e-07, 7.88371017e-07, 7.86893468e-07, 7.85988369e-07,\n",
       "       7.80400777e-07, 7.76411085e-07, 7.73502849e-07, 7.68720967e-07,\n",
       "       7.63751914e-07, 7.55493986e-07, 7.54625014e-07, 7.48169683e-07,\n",
       "       7.47410086e-07, 7.43380337e-07, 7.32084887e-07, 7.29451102e-07,\n",
       "       7.29354824e-07, 7.15766864e-07, 7.12019093e-07, 7.10452283e-07,\n",
       "       7.00034053e-07, 6.95061478e-07, 6.92365037e-07, 6.87277886e-07,\n",
       "       6.80976395e-07, 6.76748349e-07, 6.71172898e-07, 6.67471136e-07,\n",
       "       6.65272570e-07, 6.61551991e-07, 6.52071885e-07, 6.43425308e-07,\n",
       "       6.42201165e-07, 6.39661593e-07, 6.34388112e-07, 6.29878999e-07,\n",
       "       6.21621865e-07, 6.21073391e-07, 6.18634099e-07, 6.17796216e-07,\n",
       "       6.14399180e-07, 6.10587689e-07, 6.00286569e-07, 5.98647840e-07,\n",
       "       5.95246855e-07, 5.94221016e-07, 5.84781349e-07, 5.83707916e-07,\n",
       "       5.79264248e-07, 5.78338034e-07, 5.68123858e-07, 5.65928071e-07,\n",
       "       5.64024268e-07, 5.57275474e-07, 5.53016152e-07, 5.46531078e-07,\n",
       "       5.42123603e-07, 5.39233978e-07, 5.34917918e-07, 5.33545784e-07,\n",
       "       5.26142838e-07, 5.23850827e-07, 5.19986098e-07, 5.13854292e-07,\n",
       "       5.08446805e-07, 5.01811050e-07, 4.99770900e-07, 4.93866443e-07,\n",
       "       4.90561839e-07, 4.84279113e-07, 4.82856930e-07, 4.80184377e-07,\n",
       "       4.79422618e-07, 4.77599066e-07, 4.70840560e-07, 4.68841544e-07,\n",
       "       4.65812411e-07, 4.59574450e-07, 4.50965136e-07, 4.45425098e-07,\n",
       "       4.42153515e-07, 4.38003131e-07, 4.32522796e-07, 4.31363780e-07,\n",
       "       4.26385425e-07, 4.23679430e-07, 4.16645785e-07, 4.15482397e-07,\n",
       "       4.09471829e-07, 4.07600053e-07, 4.02520134e-07, 3.99100130e-07,\n",
       "       3.87660431e-07, 3.84372361e-07, 3.81326875e-07, 3.78613071e-07,\n",
       "       3.73508332e-07, 3.73051482e-07, 3.66754769e-07, 3.61691588e-07,\n",
       "       3.58464818e-07, 3.50306600e-07, 3.48276824e-07, 3.38116309e-07,\n",
       "       3.36856609e-07, 3.34548519e-07, 3.32773302e-07, 3.31504132e-07,\n",
       "       3.28780127e-07, 3.14120099e-07, 3.13492573e-07, 3.05066784e-07,\n",
       "       3.04229035e-07, 2.99065641e-07, 2.98814637e-07, 2.93516115e-07,\n",
       "       2.86997086e-07, 2.86426876e-07, 2.76023370e-07, 2.70311775e-07,\n",
       "       2.66013454e-07, 2.65653666e-07, 2.62001490e-07, 2.50823973e-07,\n",
       "       2.45167615e-07, 2.43291142e-07, 2.42662930e-07, 2.32139499e-07,\n",
       "       2.29426431e-07, 2.25648514e-07, 2.22355007e-07, 2.20516920e-07,\n",
       "       2.12374911e-07, 2.07739509e-07, 2.07404862e-07, 2.03178663e-07,\n",
       "       1.99762205e-07, 1.95673929e-07, 1.91135826e-07, 1.86693939e-07,\n",
       "       1.82568205e-07, 1.73510822e-07, 1.70724545e-07, 1.62572357e-07,\n",
       "       1.61688997e-07, 1.58838177e-07, 1.57870423e-07, 1.53433323e-07,\n",
       "       1.50592576e-07, 1.47900560e-07, 1.36496603e-07, 1.33317319e-07,\n",
       "       1.30074048e-07, 1.25140317e-07, 1.21534314e-07, 1.12379957e-07,\n",
       "       1.10496181e-07, 1.04468736e-07, 1.04468020e-07, 9.51792106e-08,\n",
       "       9.51380789e-08, 8.59939258e-08, 8.51781594e-08, 7.47553919e-08,\n",
       "       7.39788601e-08, 7.08508319e-08, 6.83526577e-08, 6.12276536e-08,\n",
       "       6.00709867e-08, 5.45758147e-08, 4.92408389e-08, 4.41025575e-08,\n",
       "       4.34835308e-08, 3.89683177e-08, 3.38756010e-08, 3.24314071e-08,\n",
       "       2.27105726e-08, 1.90492180e-08, 1.21300820e-08, 9.45965779e-09,\n",
       "       8.48541207e-09, 3.74321307e-09, 3.51903487e-09, 2.35622088e-15])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# singular values (vector)\n",
    "# is a 1D array of a’s singular values\n",
    "s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "17eacee4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(500,)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "b29dcd31",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Unitary arrays (matrix)\n",
    "# vh \n",
    "# Actually here have same result as the \"u\" "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "29ad92f8",
   "metadata": {},
   "outputs": [],
   "source": [
    "vh = pd.DataFrame(vh)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "da6cbb26",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
       "      <th>4</th>\n",
       "      <th>5</th>\n",
       "      <th>6</th>\n",
       "      <th>7</th>\n",
       "      <th>8</th>\n",
       "      <th>9</th>\n",
       "      <th>...</th>\n",
       "      <th>490</th>\n",
       "      <th>491</th>\n",
       "      <th>492</th>\n",
       "      <th>493</th>\n",
       "      <th>494</th>\n",
       "      <th>495</th>\n",
       "      <th>496</th>\n",
       "      <th>497</th>\n",
       "      <th>498</th>\n",
       "      <th>499</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-0.006387</td>\n",
       "      <td>-0.010837</td>\n",
       "      <td>-0.009057</td>\n",
       "      <td>-0.010602</td>\n",
       "      <td>-0.017506</td>\n",
       "      <td>-0.020772</td>\n",
       "      <td>0.002205</td>\n",
       "      <td>-0.007663</td>\n",
       "      <td>-0.006408</td>\n",
       "      <td>-0.021189</td>\n",
       "      <td>...</td>\n",
       "      <td>0.010054</td>\n",
       "      <td>0.018009</td>\n",
       "      <td>0.018255</td>\n",
       "      <td>0.030588</td>\n",
       "      <td>-0.014835</td>\n",
       "      <td>-0.015570</td>\n",
       "      <td>-0.003109</td>\n",
       "      <td>0.011545</td>\n",
       "      <td>-0.002565</td>\n",
       "      <td>-0.011286</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.023537</td>\n",
       "      <td>0.005218</td>\n",
       "      <td>0.006176</td>\n",
       "      <td>0.003314</td>\n",
       "      <td>0.042395</td>\n",
       "      <td>0.036854</td>\n",
       "      <td>0.018510</td>\n",
       "      <td>0.002920</td>\n",
       "      <td>0.009807</td>\n",
       "      <td>-0.021750</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.011303</td>\n",
       "      <td>-0.011063</td>\n",
       "      <td>-0.017403</td>\n",
       "      <td>-0.003624</td>\n",
       "      <td>-0.026346</td>\n",
       "      <td>-0.035488</td>\n",
       "      <td>-0.035706</td>\n",
       "      <td>-0.017246</td>\n",
       "      <td>-0.008398</td>\n",
       "      <td>-0.008489</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-0.004673</td>\n",
       "      <td>0.014919</td>\n",
       "      <td>0.027448</td>\n",
       "      <td>0.021919</td>\n",
       "      <td>0.023584</td>\n",
       "      <td>0.030860</td>\n",
       "      <td>-0.002765</td>\n",
       "      <td>0.024576</td>\n",
       "      <td>0.020967</td>\n",
       "      <td>0.101282</td>\n",
       "      <td>...</td>\n",
       "      <td>0.079818</td>\n",
       "      <td>0.086883</td>\n",
       "      <td>0.023343</td>\n",
       "      <td>-0.012814</td>\n",
       "      <td>0.033091</td>\n",
       "      <td>0.021553</td>\n",
       "      <td>0.000365</td>\n",
       "      <td>-0.037298</td>\n",
       "      <td>-0.042501</td>\n",
       "      <td>-0.023948</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.027282</td>\n",
       "      <td>-0.048530</td>\n",
       "      <td>0.044896</td>\n",
       "      <td>0.007954</td>\n",
       "      <td>0.022463</td>\n",
       "      <td>0.026459</td>\n",
       "      <td>-0.035689</td>\n",
       "      <td>-0.030301</td>\n",
       "      <td>0.013820</td>\n",
       "      <td>0.045942</td>\n",
       "      <td>...</td>\n",
       "      <td>0.024963</td>\n",
       "      <td>0.014741</td>\n",
       "      <td>-0.020335</td>\n",
       "      <td>-0.019253</td>\n",
       "      <td>0.031483</td>\n",
       "      <td>-0.036509</td>\n",
       "      <td>-0.074290</td>\n",
       "      <td>-0.015249</td>\n",
       "      <td>0.032443</td>\n",
       "      <td>0.089253</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-0.048683</td>\n",
       "      <td>-0.016666</td>\n",
       "      <td>-0.030452</td>\n",
       "      <td>-0.000284</td>\n",
       "      <td>-0.041785</td>\n",
       "      <td>-0.024583</td>\n",
       "      <td>0.020820</td>\n",
       "      <td>-0.047374</td>\n",
       "      <td>0.000348</td>\n",
       "      <td>-0.007839</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.018395</td>\n",
       "      <td>-0.045159</td>\n",
       "      <td>0.013354</td>\n",
       "      <td>-0.037162</td>\n",
       "      <td>-0.032157</td>\n",
       "      <td>0.071049</td>\n",
       "      <td>-0.008611</td>\n",
       "      <td>-0.027140</td>\n",
       "      <td>0.023240</td>\n",
       "      <td>0.056195</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>495</th>\n",
       "      <td>-0.011797</td>\n",
       "      <td>-0.070397</td>\n",
       "      <td>-0.008014</td>\n",
       "      <td>-0.041899</td>\n",
       "      <td>-0.006870</td>\n",
       "      <td>-0.010960</td>\n",
       "      <td>-0.011996</td>\n",
       "      <td>0.043915</td>\n",
       "      <td>-0.055611</td>\n",
       "      <td>0.022623</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.054625</td>\n",
       "      <td>-0.031026</td>\n",
       "      <td>0.003804</td>\n",
       "      <td>-0.090840</td>\n",
       "      <td>0.031022</td>\n",
       "      <td>-0.064265</td>\n",
       "      <td>0.085795</td>\n",
       "      <td>0.050347</td>\n",
       "      <td>0.040020</td>\n",
       "      <td>-0.022778</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>496</th>\n",
       "      <td>-0.027347</td>\n",
       "      <td>-0.067255</td>\n",
       "      <td>0.006137</td>\n",
       "      <td>-0.013406</td>\n",
       "      <td>0.038308</td>\n",
       "      <td>-0.004481</td>\n",
       "      <td>0.039573</td>\n",
       "      <td>0.007133</td>\n",
       "      <td>0.009797</td>\n",
       "      <td>-0.093589</td>\n",
       "      <td>...</td>\n",
       "      <td>0.019867</td>\n",
       "      <td>-0.044339</td>\n",
       "      <td>-0.041987</td>\n",
       "      <td>-0.069452</td>\n",
       "      <td>-0.059265</td>\n",
       "      <td>-0.015115</td>\n",
       "      <td>-0.018858</td>\n",
       "      <td>-0.031000</td>\n",
       "      <td>0.013491</td>\n",
       "      <td>-0.001411</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>497</th>\n",
       "      <td>-0.005959</td>\n",
       "      <td>0.004390</td>\n",
       "      <td>0.099489</td>\n",
       "      <td>0.034253</td>\n",
       "      <td>-0.023583</td>\n",
       "      <td>0.028885</td>\n",
       "      <td>0.028907</td>\n",
       "      <td>-0.023617</td>\n",
       "      <td>0.042867</td>\n",
       "      <td>-0.032751</td>\n",
       "      <td>...</td>\n",
       "      <td>0.055534</td>\n",
       "      <td>0.009855</td>\n",
       "      <td>-0.038935</td>\n",
       "      <td>0.092464</td>\n",
       "      <td>0.001123</td>\n",
       "      <td>0.079277</td>\n",
       "      <td>0.029559</td>\n",
       "      <td>0.001908</td>\n",
       "      <td>0.037706</td>\n",
       "      <td>-0.057886</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>498</th>\n",
       "      <td>-0.039187</td>\n",
       "      <td>0.052487</td>\n",
       "      <td>0.025765</td>\n",
       "      <td>0.014198</td>\n",
       "      <td>0.044797</td>\n",
       "      <td>0.025664</td>\n",
       "      <td>-0.032950</td>\n",
       "      <td>0.007805</td>\n",
       "      <td>-0.028169</td>\n",
       "      <td>0.083690</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.028993</td>\n",
       "      <td>0.027184</td>\n",
       "      <td>0.035850</td>\n",
       "      <td>0.011074</td>\n",
       "      <td>0.056673</td>\n",
       "      <td>0.016957</td>\n",
       "      <td>0.017190</td>\n",
       "      <td>-0.065878</td>\n",
       "      <td>-0.037748</td>\n",
       "      <td>-0.003453</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>499</th>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>500 rows × 500 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "          0         1         2         3         4         5         6    \\\n",
       "0   -0.006387 -0.010837 -0.009057 -0.010602 -0.017506 -0.020772  0.002205   \n",
       "1    0.023537  0.005218  0.006176  0.003314  0.042395  0.036854  0.018510   \n",
       "2   -0.004673  0.014919  0.027448  0.021919  0.023584  0.030860 -0.002765   \n",
       "3    0.027282 -0.048530  0.044896  0.007954  0.022463  0.026459 -0.035689   \n",
       "4   -0.048683 -0.016666 -0.030452 -0.000284 -0.041785 -0.024583  0.020820   \n",
       "..        ...       ...       ...       ...       ...       ...       ...   \n",
       "495 -0.011797 -0.070397 -0.008014 -0.041899 -0.006870 -0.010960 -0.011996   \n",
       "496 -0.027347 -0.067255  0.006137 -0.013406  0.038308 -0.004481  0.039573   \n",
       "497 -0.005959  0.004390  0.099489  0.034253 -0.023583  0.028885  0.028907   \n",
       "498 -0.039187  0.052487  0.025765  0.014198  0.044797  0.025664 -0.032950   \n",
       "499 -0.044721 -0.044721 -0.044721 -0.044721 -0.044721 -0.044721 -0.044721   \n",
       "\n",
       "          7         8         9    ...       490       491       492  \\\n",
       "0   -0.007663 -0.006408 -0.021189  ...  0.010054  0.018009  0.018255   \n",
       "1    0.002920  0.009807 -0.021750  ... -0.011303 -0.011063 -0.017403   \n",
       "2    0.024576  0.020967  0.101282  ...  0.079818  0.086883  0.023343   \n",
       "3   -0.030301  0.013820  0.045942  ...  0.024963  0.014741 -0.020335   \n",
       "4   -0.047374  0.000348 -0.007839  ... -0.018395 -0.045159  0.013354   \n",
       "..        ...       ...       ...  ...       ...       ...       ...   \n",
       "495  0.043915 -0.055611  0.022623  ... -0.054625 -0.031026  0.003804   \n",
       "496  0.007133  0.009797 -0.093589  ...  0.019867 -0.044339 -0.041987   \n",
       "497 -0.023617  0.042867 -0.032751  ...  0.055534  0.009855 -0.038935   \n",
       "498  0.007805 -0.028169  0.083690  ... -0.028993  0.027184  0.035850   \n",
       "499 -0.044721 -0.044721 -0.044721  ... -0.044721 -0.044721 -0.044721   \n",
       "\n",
       "          493       494       495       496       497       498       499  \n",
       "0    0.030588 -0.014835 -0.015570 -0.003109  0.011545 -0.002565 -0.011286  \n",
       "1   -0.003624 -0.026346 -0.035488 -0.035706 -0.017246 -0.008398 -0.008489  \n",
       "2   -0.012814  0.033091  0.021553  0.000365 -0.037298 -0.042501 -0.023948  \n",
       "3   -0.019253  0.031483 -0.036509 -0.074290 -0.015249  0.032443  0.089253  \n",
       "4   -0.037162 -0.032157  0.071049 -0.008611 -0.027140  0.023240  0.056195  \n",
       "..        ...       ...       ...       ...       ...       ...       ...  \n",
       "495 -0.090840  0.031022 -0.064265  0.085795  0.050347  0.040020 -0.022778  \n",
       "496 -0.069452 -0.059265 -0.015115 -0.018858 -0.031000  0.013491 -0.001411  \n",
       "497  0.092464  0.001123  0.079277  0.029559  0.001908  0.037706 -0.057886  \n",
       "498  0.011074  0.056673  0.016957  0.017190 -0.065878 -0.037748 -0.003453  \n",
       "499 -0.044721 -0.044721 -0.044721 -0.044721 -0.044721 -0.044721 -0.044721  \n",
       "\n",
       "[500 rows x 500 columns]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vh"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "728740b8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(500, 500)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vh.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "2eab7d8e",
   "metadata": {},
   "outputs": [],
   "source": [
    "# LAMBDA\n",
    "lambda_ = s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "a3428a24",
   "metadata": {},
   "outputs": [],
   "source": [
    "cumCntrb = np.cumsum(lambda_/len(lambda_))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "d6807745",
   "metadata": {},
   "outputs": [],
   "source": [
    "lambd = lambda_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "8d0d10a4",
   "metadata": {},
   "outputs": [],
   "source": [
    "lambd = pd.DataFrame(lambd)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "fdefb406",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
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       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>0</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>7.415728e+00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>4.877160e+00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>2.917023e+00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>2.263043e+00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>2.091741e+00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>495</th>\n",
       "      <td>9.459658e-09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>496</th>\n",
       "      <td>8.485412e-09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>497</th>\n",
       "      <td>3.743213e-09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>498</th>\n",
       "      <td>3.519035e-09</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>499</th>\n",
       "      <td>2.356221e-15</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>500 rows × 1 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                0\n",
       "0    7.415728e+00\n",
       "1    4.877160e+00\n",
       "2    2.917023e+00\n",
       "3    2.263043e+00\n",
       "4    2.091741e+00\n",
       "..            ...\n",
       "495  9.459658e-09\n",
       "496  8.485412e-09\n",
       "497  3.743213e-09\n",
       "498  3.519035e-09\n",
       "499  2.356221e-15\n",
       "\n",
       "[500 rows x 1 columns]"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lambd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "b3a2273d",
   "metadata": {},
   "outputs": [],
   "source": [
    "# COEFFICIENT\n",
    "a = np.sqrt(len(df_scl)*lambd)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "f8651e9e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(500,)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# a\n",
    "a.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "04544c86",
   "metadata": {},
   "outputs": [],
   "source": [
    "coefficient = np.divide(u, a.T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "dbc9ff64",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(500, 500)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coefficient.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "f60290cc",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
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       "      <th></th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
       "      <th>4</th>\n",
       "      <th>5</th>\n",
       "      <th>6</th>\n",
       "      <th>7</th>\n",
       "      <th>8</th>\n",
       "      <th>9</th>\n",
       "      <th>...</th>\n",
       "      <th>490</th>\n",
       "      <th>491</th>\n",
       "      <th>492</th>\n",
       "      <th>493</th>\n",
       "      <th>494</th>\n",
       "      <th>495</th>\n",
       "      <th>496</th>\n",
       "      <th>497</th>\n",
       "      <th>498</th>\n",
       "      <th>499</th>\n",
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       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-0.000105</td>\n",
       "      <td>0.000477</td>\n",
       "      <td>-0.000122</td>\n",
       "      <td>0.000811</td>\n",
       "      <td>-0.001505</td>\n",
       "      <td>0.000063</td>\n",
       "      <td>-0.001167</td>\n",
       "      <td>0.000629</td>\n",
       "      <td>-0.000325</td>\n",
       "      <td>0.001204</td>\n",
       "      <td>...</td>\n",
       "      <td>-12.376775</td>\n",
       "      <td>10.382078</td>\n",
       "      <td>-12.884696</td>\n",
       "      <td>1.690135</td>\n",
       "      <td>-15.087133</td>\n",
       "      <td>5.424561</td>\n",
       "      <td>-13.276635</td>\n",
       "      <td>4.355762</td>\n",
       "      <td>-29.542362</td>\n",
       "      <td>-41202.373439</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-0.000178</td>\n",
       "      <td>0.000106</td>\n",
       "      <td>0.000391</td>\n",
       "      <td>-0.001443</td>\n",
       "      <td>-0.000515</td>\n",
       "      <td>0.000089</td>\n",
       "      <td>0.000312</td>\n",
       "      <td>-0.000700</td>\n",
       "      <td>0.001253</td>\n",
       "      <td>-0.001811</td>\n",
       "      <td>...</td>\n",
       "      <td>-12.772763</td>\n",
       "      <td>-0.766002</td>\n",
       "      <td>-4.632076</td>\n",
       "      <td>-26.671157</td>\n",
       "      <td>17.284659</td>\n",
       "      <td>32.369348</td>\n",
       "      <td>-32.651312</td>\n",
       "      <td>-3.208731</td>\n",
       "      <td>39.569108</td>\n",
       "      <td>-41202.330175</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-0.000149</td>\n",
       "      <td>0.000125</td>\n",
       "      <td>0.000719</td>\n",
       "      <td>0.001335</td>\n",
       "      <td>-0.000942</td>\n",
       "      <td>-0.002309</td>\n",
       "      <td>0.000356</td>\n",
       "      <td>-0.000370</td>\n",
       "      <td>0.001192</td>\n",
       "      <td>-0.001477</td>\n",
       "      <td>...</td>\n",
       "      <td>1.126346</td>\n",
       "      <td>-11.511614</td>\n",
       "      <td>-7.075241</td>\n",
       "      <td>-7.883159</td>\n",
       "      <td>16.271328</td>\n",
       "      <td>3.685136</td>\n",
       "      <td>2.979576</td>\n",
       "      <td>-72.722020</td>\n",
       "      <td>19.423568</td>\n",
       "      <td>-41202.344088</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-0.000174</td>\n",
       "      <td>0.000067</td>\n",
       "      <td>0.000574</td>\n",
       "      <td>0.000236</td>\n",
       "      <td>-0.000009</td>\n",
       "      <td>-0.000804</td>\n",
       "      <td>0.000018</td>\n",
       "      <td>0.000073</td>\n",
       "      <td>0.002019</td>\n",
       "      <td>-0.002369</td>\n",
       "      <td>...</td>\n",
       "      <td>0.266576</td>\n",
       "      <td>5.497336</td>\n",
       "      <td>-4.848839</td>\n",
       "      <td>16.423661</td>\n",
       "      <td>-33.780983</td>\n",
       "      <td>19.265623</td>\n",
       "      <td>-6.508220</td>\n",
       "      <td>-25.037732</td>\n",
       "      <td>10.703982</td>\n",
       "      <td>-41202.350829</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-0.000287</td>\n",
       "      <td>0.000859</td>\n",
       "      <td>0.000618</td>\n",
       "      <td>0.000668</td>\n",
       "      <td>-0.001292</td>\n",
       "      <td>0.001648</td>\n",
       "      <td>-0.001266</td>\n",
       "      <td>0.001282</td>\n",
       "      <td>-0.001864</td>\n",
       "      <td>-0.000639</td>\n",
       "      <td>...</td>\n",
       "      <td>1.164543</td>\n",
       "      <td>-25.253771</td>\n",
       "      <td>-11.146677</td>\n",
       "      <td>12.290213</td>\n",
       "      <td>-39.865854</td>\n",
       "      <td>3.158824</td>\n",
       "      <td>18.598059</td>\n",
       "      <td>17.238070</td>\n",
       "      <td>33.771808</td>\n",
       "      <td>-41202.331113</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>495</th>\n",
       "      <td>-0.000256</td>\n",
       "      <td>-0.000719</td>\n",
       "      <td>0.000564</td>\n",
       "      <td>-0.001085</td>\n",
       "      <td>0.002197</td>\n",
       "      <td>0.001235</td>\n",
       "      <td>0.002049</td>\n",
       "      <td>0.000149</td>\n",
       "      <td>0.001369</td>\n",
       "      <td>-0.003077</td>\n",
       "      <td>...</td>\n",
       "      <td>-9.216399</td>\n",
       "      <td>1.101597</td>\n",
       "      <td>-5.846034</td>\n",
       "      <td>-6.865274</td>\n",
       "      <td>11.673180</td>\n",
       "      <td>29.549571</td>\n",
       "      <td>-7.338150</td>\n",
       "      <td>-57.948372</td>\n",
       "      <td>12.783220</td>\n",
       "      <td>-41202.347740</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>496</th>\n",
       "      <td>-0.000051</td>\n",
       "      <td>-0.000723</td>\n",
       "      <td>0.000010</td>\n",
       "      <td>-0.002209</td>\n",
       "      <td>-0.000266</td>\n",
       "      <td>-0.001866</td>\n",
       "      <td>-0.001185</td>\n",
       "      <td>-0.001148</td>\n",
       "      <td>0.000779</td>\n",
       "      <td>-0.000153</td>\n",
       "      <td>...</td>\n",
       "      <td>-9.129216</td>\n",
       "      <td>-5.903220</td>\n",
       "      <td>16.769751</td>\n",
       "      <td>-2.298028</td>\n",
       "      <td>-28.239895</td>\n",
       "      <td>-39.449170</td>\n",
       "      <td>-9.155486</td>\n",
       "      <td>-21.606605</td>\n",
       "      <td>12.958875</td>\n",
       "      <td>-41202.351652</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>497</th>\n",
       "      <td>0.000190</td>\n",
       "      <td>-0.000349</td>\n",
       "      <td>-0.000977</td>\n",
       "      <td>-0.000453</td>\n",
       "      <td>-0.000839</td>\n",
       "      <td>-0.001562</td>\n",
       "      <td>-0.003574</td>\n",
       "      <td>0.000513</td>\n",
       "      <td>-0.001148</td>\n",
       "      <td>0.000184</td>\n",
       "      <td>...</td>\n",
       "      <td>6.469670</td>\n",
       "      <td>-3.966850</td>\n",
       "      <td>-8.414350</td>\n",
       "      <td>17.519548</td>\n",
       "      <td>25.051070</td>\n",
       "      <td>-23.149800</td>\n",
       "      <td>-15.050310</td>\n",
       "      <td>-1.394425</td>\n",
       "      <td>-49.664138</td>\n",
       "      <td>-41202.382560</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>498</th>\n",
       "      <td>-0.000042</td>\n",
       "      <td>-0.000170</td>\n",
       "      <td>-0.001113</td>\n",
       "      <td>0.000964</td>\n",
       "      <td>0.000719</td>\n",
       "      <td>-0.000604</td>\n",
       "      <td>-0.000384</td>\n",
       "      <td>-0.000127</td>\n",
       "      <td>-0.002323</td>\n",
       "      <td>0.000124</td>\n",
       "      <td>...</td>\n",
       "      <td>-16.393860</td>\n",
       "      <td>6.711271</td>\n",
       "      <td>13.039767</td>\n",
       "      <td>-7.341459</td>\n",
       "      <td>-18.191699</td>\n",
       "      <td>-18.401514</td>\n",
       "      <td>6.549914</td>\n",
       "      <td>-27.561371</td>\n",
       "      <td>-28.457365</td>\n",
       "      <td>-41202.371496</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>499</th>\n",
       "      <td>-0.000185</td>\n",
       "      <td>-0.000172</td>\n",
       "      <td>-0.000627</td>\n",
       "      <td>0.002653</td>\n",
       "      <td>0.001738</td>\n",
       "      <td>0.000827</td>\n",
       "      <td>-0.001890</td>\n",
       "      <td>0.000525</td>\n",
       "      <td>-0.000484</td>\n",
       "      <td>-0.001028</td>\n",
       "      <td>...</td>\n",
       "      <td>19.345853</td>\n",
       "      <td>-16.892312</td>\n",
       "      <td>-21.787986</td>\n",
       "      <td>-10.671797</td>\n",
       "      <td>10.492002</td>\n",
       "      <td>10.473404</td>\n",
       "      <td>-0.685159</td>\n",
       "      <td>42.312393</td>\n",
       "      <td>-2.603216</td>\n",
       "      <td>-41202.348600</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>500 rows × 500 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "          0         1         2         3         4         5         6    \\\n",
       "0   -0.000105  0.000477 -0.000122  0.000811 -0.001505  0.000063 -0.001167   \n",
       "1   -0.000178  0.000106  0.000391 -0.001443 -0.000515  0.000089  0.000312   \n",
       "2   -0.000149  0.000125  0.000719  0.001335 -0.000942 -0.002309  0.000356   \n",
       "3   -0.000174  0.000067  0.000574  0.000236 -0.000009 -0.000804  0.000018   \n",
       "4   -0.000287  0.000859  0.000618  0.000668 -0.001292  0.001648 -0.001266   \n",
       "..        ...       ...       ...       ...       ...       ...       ...   \n",
       "495 -0.000256 -0.000719  0.000564 -0.001085  0.002197  0.001235  0.002049   \n",
       "496 -0.000051 -0.000723  0.000010 -0.002209 -0.000266 -0.001866 -0.001185   \n",
       "497  0.000190 -0.000349 -0.000977 -0.000453 -0.000839 -0.001562 -0.003574   \n",
       "498 -0.000042 -0.000170 -0.001113  0.000964  0.000719 -0.000604 -0.000384   \n",
       "499 -0.000185 -0.000172 -0.000627  0.002653  0.001738  0.000827 -0.001890   \n",
       "\n",
       "          7         8         9    ...        490        491        492  \\\n",
       "0    0.000629 -0.000325  0.001204  ... -12.376775  10.382078 -12.884696   \n",
       "1   -0.000700  0.001253 -0.001811  ... -12.772763  -0.766002  -4.632076   \n",
       "2   -0.000370  0.001192 -0.001477  ...   1.126346 -11.511614  -7.075241   \n",
       "3    0.000073  0.002019 -0.002369  ...   0.266576   5.497336  -4.848839   \n",
       "4    0.001282 -0.001864 -0.000639  ...   1.164543 -25.253771 -11.146677   \n",
       "..        ...       ...       ...  ...        ...        ...        ...   \n",
       "495  0.000149  0.001369 -0.003077  ...  -9.216399   1.101597  -5.846034   \n",
       "496 -0.001148  0.000779 -0.000153  ...  -9.129216  -5.903220  16.769751   \n",
       "497  0.000513 -0.001148  0.000184  ...   6.469670  -3.966850  -8.414350   \n",
       "498 -0.000127 -0.002323  0.000124  ... -16.393860   6.711271  13.039767   \n",
       "499  0.000525 -0.000484 -0.001028  ...  19.345853 -16.892312 -21.787986   \n",
       "\n",
       "           493        494        495        496        497        498  \\\n",
       "0     1.690135 -15.087133   5.424561 -13.276635   4.355762 -29.542362   \n",
       "1   -26.671157  17.284659  32.369348 -32.651312  -3.208731  39.569108   \n",
       "2    -7.883159  16.271328   3.685136   2.979576 -72.722020  19.423568   \n",
       "3    16.423661 -33.780983  19.265623  -6.508220 -25.037732  10.703982   \n",
       "4    12.290213 -39.865854   3.158824  18.598059  17.238070  33.771808   \n",
       "..         ...        ...        ...        ...        ...        ...   \n",
       "495  -6.865274  11.673180  29.549571  -7.338150 -57.948372  12.783220   \n",
       "496  -2.298028 -28.239895 -39.449170  -9.155486 -21.606605  12.958875   \n",
       "497  17.519548  25.051070 -23.149800 -15.050310  -1.394425 -49.664138   \n",
       "498  -7.341459 -18.191699 -18.401514   6.549914 -27.561371 -28.457365   \n",
       "499 -10.671797  10.492002  10.473404  -0.685159  42.312393  -2.603216   \n",
       "\n",
       "              499  \n",
       "0   -41202.373439  \n",
       "1   -41202.330175  \n",
       "2   -41202.344088  \n",
       "3   -41202.350829  \n",
       "4   -41202.331113  \n",
       "..            ...  \n",
       "495 -41202.347740  \n",
       "496 -41202.351652  \n",
       "497 -41202.382560  \n",
       "498 -41202.371496  \n",
       "499 -41202.348600  \n",
       "\n",
       "[500 rows x 500 columns]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coefficient"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b77c3701",
   "metadata": {},
   "source": [
    "## Scipy SVD"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "59340692",
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.linalg import svd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "1df973a7",
   "metadata": {},
   "outputs": [],
   "source": [
    "U, s, VT = svd(K_c/len(df_scl))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "410ff19d",
   "metadata": {},
   "outputs": [],
   "source": [
    "U = pd.DataFrame(U)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "4bbea167",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
       "      <th>4</th>\n",
       "      <th>5</th>\n",
       "      <th>6</th>\n",
       "      <th>7</th>\n",
       "      <th>8</th>\n",
       "      <th>9</th>\n",
       "      <th>...</th>\n",
       "      <th>490</th>\n",
       "      <th>491</th>\n",
       "      <th>492</th>\n",
       "      <th>493</th>\n",
       "      <th>494</th>\n",
       "      <th>495</th>\n",
       "      <th>496</th>\n",
       "      <th>497</th>\n",
       "      <th>498</th>\n",
       "      <th>499</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-0.006387</td>\n",
       "      <td>0.023537</td>\n",
       "      <td>-0.004673</td>\n",
       "      <td>0.027282</td>\n",
       "      <td>-0.048683</td>\n",
       "      <td>0.001987</td>\n",
       "      <td>-0.036125</td>\n",
       "      <td>0.017959</td>\n",
       "      <td>-0.008994</td>\n",
       "      <td>0.032167</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.050937</td>\n",
       "      <td>0.041807</td>\n",
       "      <td>-0.043418</td>\n",
       "      <td>0.005216</td>\n",
       "      <td>-0.037156</td>\n",
       "      <td>0.011797</td>\n",
       "      <td>-0.027347</td>\n",
       "      <td>0.005959</td>\n",
       "      <td>-0.039187</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-0.010837</td>\n",
       "      <td>0.005218</td>\n",
       "      <td>0.014919</td>\n",
       "      <td>-0.048530</td>\n",
       "      <td>-0.016666</td>\n",
       "      <td>0.002775</td>\n",
       "      <td>0.009661</td>\n",
       "      <td>-0.019979</td>\n",
       "      <td>0.034658</td>\n",
       "      <td>-0.048371</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.052567</td>\n",
       "      <td>-0.003085</td>\n",
       "      <td>-0.015609</td>\n",
       "      <td>-0.082312</td>\n",
       "      <td>0.042567</td>\n",
       "      <td>0.070397</td>\n",
       "      <td>-0.067255</td>\n",
       "      <td>-0.004390</td>\n",
       "      <td>0.052487</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-0.009057</td>\n",
       "      <td>0.006176</td>\n",
       "      <td>0.027448</td>\n",
       "      <td>0.044896</td>\n",
       "      <td>-0.030452</td>\n",
       "      <td>-0.072360</td>\n",
       "      <td>0.011022</td>\n",
       "      <td>-0.010565</td>\n",
       "      <td>0.032952</td>\n",
       "      <td>-0.039450</td>\n",
       "      <td>...</td>\n",
       "      <td>0.004636</td>\n",
       "      <td>-0.046356</td>\n",
       "      <td>-0.023842</td>\n",
       "      <td>-0.024329</td>\n",
       "      <td>0.040072</td>\n",
       "      <td>0.008014</td>\n",
       "      <td>0.006137</td>\n",
       "      <td>-0.099489</td>\n",
       "      <td>0.025765</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-0.010602</td>\n",
       "      <td>0.003314</td>\n",
       "      <td>0.021919</td>\n",
       "      <td>0.007954</td>\n",
       "      <td>-0.000284</td>\n",
       "      <td>-0.025181</td>\n",
       "      <td>0.000550</td>\n",
       "      <td>0.002075</td>\n",
       "      <td>0.055837</td>\n",
       "      <td>-0.063263</td>\n",
       "      <td>...</td>\n",
       "      <td>0.001097</td>\n",
       "      <td>0.022137</td>\n",
       "      <td>-0.016339</td>\n",
       "      <td>0.050687</td>\n",
       "      <td>-0.083193</td>\n",
       "      <td>0.041899</td>\n",
       "      <td>-0.013406</td>\n",
       "      <td>-0.034253</td>\n",
       "      <td>0.014198</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-0.017506</td>\n",
       "      <td>0.042395</td>\n",
       "      <td>0.023584</td>\n",
       "      <td>0.022463</td>\n",
       "      <td>-0.041785</td>\n",
       "      <td>0.051631</td>\n",
       "      <td>-0.039164</td>\n",
       "      <td>0.036588</td>\n",
       "      <td>-0.051554</td>\n",
       "      <td>-0.017074</td>\n",
       "      <td>...</td>\n",
       "      <td>0.004793</td>\n",
       "      <td>-0.101694</td>\n",
       "      <td>-0.037562</td>\n",
       "      <td>0.037930</td>\n",
       "      <td>-0.098179</td>\n",
       "      <td>0.006870</td>\n",
       "      <td>0.038308</td>\n",
       "      <td>0.023583</td>\n",
       "      <td>0.044797</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>495</th>\n",
       "      <td>-0.015570</td>\n",
       "      <td>-0.035488</td>\n",
       "      <td>0.021553</td>\n",
       "      <td>-0.036509</td>\n",
       "      <td>0.071049</td>\n",
       "      <td>0.038697</td>\n",
       "      <td>0.063405</td>\n",
       "      <td>0.004241</td>\n",
       "      <td>0.037852</td>\n",
       "      <td>-0.082169</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.037931</td>\n",
       "      <td>0.004436</td>\n",
       "      <td>-0.019700</td>\n",
       "      <td>-0.021188</td>\n",
       "      <td>0.028748</td>\n",
       "      <td>0.064265</td>\n",
       "      <td>-0.015115</td>\n",
       "      <td>-0.079277</td>\n",
       "      <td>0.016957</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>496</th>\n",
       "      <td>-0.003109</td>\n",
       "      <td>-0.035706</td>\n",
       "      <td>0.000365</td>\n",
       "      <td>-0.074290</td>\n",
       "      <td>-0.008611</td>\n",
       "      <td>-0.058478</td>\n",
       "      <td>-0.036679</td>\n",
       "      <td>-0.032770</td>\n",
       "      <td>0.021545</td>\n",
       "      <td>-0.004079</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.037572</td>\n",
       "      <td>-0.023772</td>\n",
       "      <td>0.056510</td>\n",
       "      <td>-0.007092</td>\n",
       "      <td>-0.069547</td>\n",
       "      <td>-0.085795</td>\n",
       "      <td>-0.018858</td>\n",
       "      <td>-0.029559</td>\n",
       "      <td>0.017190</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>497</th>\n",
       "      <td>0.011545</td>\n",
       "      <td>-0.017246</td>\n",
       "      <td>-0.037298</td>\n",
       "      <td>-0.015249</td>\n",
       "      <td>-0.027140</td>\n",
       "      <td>-0.048940</td>\n",
       "      <td>-0.110584</td>\n",
       "      <td>0.014648</td>\n",
       "      <td>-0.031744</td>\n",
       "      <td>0.004925</td>\n",
       "      <td>...</td>\n",
       "      <td>0.026626</td>\n",
       "      <td>-0.015974</td>\n",
       "      <td>-0.028354</td>\n",
       "      <td>0.054069</td>\n",
       "      <td>0.061694</td>\n",
       "      <td>-0.050347</td>\n",
       "      <td>-0.031000</td>\n",
       "      <td>-0.001908</td>\n",
       "      <td>-0.065878</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>498</th>\n",
       "      <td>-0.002565</td>\n",
       "      <td>-0.008398</td>\n",
       "      <td>-0.042501</td>\n",
       "      <td>0.032443</td>\n",
       "      <td>0.023240</td>\n",
       "      <td>-0.018927</td>\n",
       "      <td>-0.011876</td>\n",
       "      <td>-0.003614</td>\n",
       "      <td>-0.064240</td>\n",
       "      <td>0.003310</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.067470</td>\n",
       "      <td>0.027025</td>\n",
       "      <td>0.043941</td>\n",
       "      <td>-0.022657</td>\n",
       "      <td>-0.044801</td>\n",
       "      <td>-0.040020</td>\n",
       "      <td>0.013491</td>\n",
       "      <td>-0.037706</td>\n",
       "      <td>-0.037748</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>499</th>\n",
       "      <td>-0.011286</td>\n",
       "      <td>-0.008489</td>\n",
       "      <td>-0.023948</td>\n",
       "      <td>0.089253</td>\n",
       "      <td>0.056195</td>\n",
       "      <td>0.025926</td>\n",
       "      <td>-0.058476</td>\n",
       "      <td>0.014983</td>\n",
       "      <td>-0.013383</td>\n",
       "      <td>-0.027448</td>\n",
       "      <td>...</td>\n",
       "      <td>0.079619</td>\n",
       "      <td>-0.068023</td>\n",
       "      <td>-0.073420</td>\n",
       "      <td>-0.032935</td>\n",
       "      <td>0.025839</td>\n",
       "      <td>0.022778</td>\n",
       "      <td>-0.001411</td>\n",
       "      <td>0.057886</td>\n",
       "      <td>-0.003453</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>500 rows × 500 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "          0         1         2         3         4         5         6    \\\n",
       "0   -0.006387  0.023537 -0.004673  0.027282 -0.048683  0.001987 -0.036125   \n",
       "1   -0.010837  0.005218  0.014919 -0.048530 -0.016666  0.002775  0.009661   \n",
       "2   -0.009057  0.006176  0.027448  0.044896 -0.030452 -0.072360  0.011022   \n",
       "3   -0.010602  0.003314  0.021919  0.007954 -0.000284 -0.025181  0.000550   \n",
       "4   -0.017506  0.042395  0.023584  0.022463 -0.041785  0.051631 -0.039164   \n",
       "..        ...       ...       ...       ...       ...       ...       ...   \n",
       "495 -0.015570 -0.035488  0.021553 -0.036509  0.071049  0.038697  0.063405   \n",
       "496 -0.003109 -0.035706  0.000365 -0.074290 -0.008611 -0.058478 -0.036679   \n",
       "497  0.011545 -0.017246 -0.037298 -0.015249 -0.027140 -0.048940 -0.110584   \n",
       "498 -0.002565 -0.008398 -0.042501  0.032443  0.023240 -0.018927 -0.011876   \n",
       "499 -0.011286 -0.008489 -0.023948  0.089253  0.056195  0.025926 -0.058476   \n",
       "\n",
       "          7         8         9    ...       490       491       492  \\\n",
       "0    0.017959 -0.008994  0.032167  ... -0.050937  0.041807 -0.043418   \n",
       "1   -0.019979  0.034658 -0.048371  ... -0.052567 -0.003085 -0.015609   \n",
       "2   -0.010565  0.032952 -0.039450  ...  0.004636 -0.046356 -0.023842   \n",
       "3    0.002075  0.055837 -0.063263  ...  0.001097  0.022137 -0.016339   \n",
       "4    0.036588 -0.051554 -0.017074  ...  0.004793 -0.101694 -0.037562   \n",
       "..        ...       ...       ...  ...       ...       ...       ...   \n",
       "495  0.004241  0.037852 -0.082169  ... -0.037931  0.004436 -0.019700   \n",
       "496 -0.032770  0.021545 -0.004079  ... -0.037572 -0.023772  0.056510   \n",
       "497  0.014648 -0.031744  0.004925  ...  0.026626 -0.015974 -0.028354   \n",
       "498 -0.003614 -0.064240  0.003310  ... -0.067470  0.027025  0.043941   \n",
       "499  0.014983 -0.013383 -0.027448  ...  0.079619 -0.068023 -0.073420   \n",
       "\n",
       "          493       494       495       496       497       498       499  \n",
       "0    0.005216 -0.037156  0.011797 -0.027347  0.005959 -0.039187 -0.044721  \n",
       "1   -0.082312  0.042567  0.070397 -0.067255 -0.004390  0.052487 -0.044721  \n",
       "2   -0.024329  0.040072  0.008014  0.006137 -0.099489  0.025765 -0.044721  \n",
       "3    0.050687 -0.083193  0.041899 -0.013406 -0.034253  0.014198 -0.044721  \n",
       "4    0.037930 -0.098179  0.006870  0.038308  0.023583  0.044797 -0.044721  \n",
       "..        ...       ...       ...       ...       ...       ...       ...  \n",
       "495 -0.021188  0.028748  0.064265 -0.015115 -0.079277  0.016957 -0.044721  \n",
       "496 -0.007092 -0.069547 -0.085795 -0.018858 -0.029559  0.017190 -0.044721  \n",
       "497  0.054069  0.061694 -0.050347 -0.031000 -0.001908 -0.065878 -0.044721  \n",
       "498 -0.022657 -0.044801 -0.040020  0.013491 -0.037706 -0.037748 -0.044721  \n",
       "499 -0.032935  0.025839  0.022778 -0.001411  0.057886 -0.003453 -0.044721  \n",
       "\n",
       "[500 rows x 500 columns]"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "746321ea",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([7.41572790e+00, 4.87716031e+00, 2.91702285e+00, 2.26304325e+00,\n",
       "       2.09174059e+00, 1.96410904e+00, 1.91524406e+00, 1.62854249e+00,\n",
       "       1.52950929e+00, 1.42664381e+00, 1.32478768e+00, 1.27554390e+00,\n",
       "       1.21253926e+00, 1.15699224e+00, 1.14223561e+00, 1.08140859e+00,\n",
       "       1.05831636e+00, 1.04167785e+00, 1.00857664e+00, 9.38895349e-01,\n",
       "       9.11099176e-01, 8.55166277e-01, 8.19187164e-01, 7.94319149e-01,\n",
       "       7.62904166e-01, 7.55881376e-01, 7.15453836e-01, 6.93704506e-01,\n",
       "       6.81613005e-01, 6.28553779e-01, 6.20353194e-01, 5.66408946e-01,\n",
       "       5.48157595e-01, 5.05362928e-01, 4.81325902e-01, 4.50461328e-01,\n",
       "       4.06325635e-01, 3.74823658e-01, 3.45686134e-01, 2.72181785e-01,\n",
       "       2.30627815e-01, 1.44735048e-01, 7.50311976e-02, 4.57077066e-02,\n",
       "       2.07157442e-02, 1.18287831e-02, 6.43691242e-03, 3.91805002e-03,\n",
       "       2.17377496e-03, 8.96794862e-05, 2.45714031e-06, 2.43711394e-06,\n",
       "       2.39359459e-06, 2.37693907e-06, 2.35836338e-06, 2.34921429e-06,\n",
       "       2.33724866e-06, 2.31192126e-06, 2.30232768e-06, 2.29175972e-06,\n",
       "       2.28768589e-06, 2.25931246e-06, 2.25001742e-06, 2.24715395e-06,\n",
       "       2.23014168e-06, 2.22951284e-06, 2.21093561e-06, 2.20820561e-06,\n",
       "       2.18339721e-06, 2.18008319e-06, 2.17625681e-06, 2.16508808e-06,\n",
       "       2.14958523e-06, 2.14923370e-06, 2.12695387e-06, 2.12111467e-06,\n",
       "       2.11449733e-06, 2.11343407e-06, 2.10549075e-06, 2.07697469e-06,\n",
       "       2.07280917e-06, 2.07216415e-06, 2.06774902e-06, 2.06594050e-06,\n",
       "       2.06061598e-06, 2.04956910e-06, 2.04566069e-06, 2.03884480e-06,\n",
       "       2.03842798e-06, 2.02565534e-06, 2.00509118e-06, 2.00317179e-06,\n",
       "       1.98763632e-06, 1.98452155e-06, 1.97803487e-06, 1.97026150e-06,\n",
       "       1.95480320e-06, 1.95122440e-06, 1.94439352e-06, 1.93567914e-06,\n",
       "       1.93236963e-06, 1.93003083e-06, 1.92401038e-06, 1.91673152e-06,\n",
       "       1.90157967e-06, 1.89286958e-06, 1.88965133e-06, 1.87702581e-06,\n",
       "       1.87108149e-06, 1.86622078e-06, 1.85837650e-06, 1.85071269e-06,\n",
       "       1.84598762e-06, 1.83882319e-06, 1.83643024e-06, 1.83057123e-06,\n",
       "       1.83029459e-06, 1.81862365e-06, 1.80927680e-06, 1.80010115e-06,\n",
       "       1.79608888e-06, 1.78684451e-06, 1.77966938e-06, 1.77246896e-06,\n",
       "       1.76787227e-06, 1.76640311e-06, 1.75784663e-06, 1.75390082e-06,\n",
       "       1.75027260e-06, 1.74440783e-06, 1.73767665e-06, 1.73484740e-06,\n",
       "       1.73197554e-06, 1.72949077e-06, 1.72351068e-06, 1.71496376e-06,\n",
       "       1.70320718e-06, 1.68753183e-06, 1.67862282e-06, 1.67801582e-06,\n",
       "       1.67081527e-06, 1.67060939e-06, 1.65720664e-06, 1.65708701e-06,\n",
       "       1.64560456e-06, 1.64476471e-06, 1.64004596e-06, 1.63131880e-06,\n",
       "       1.62894734e-06, 1.62315442e-06, 1.61195611e-06, 1.60366643e-06,\n",
       "       1.59727483e-06, 1.59040509e-06, 1.58651732e-06, 1.58319003e-06,\n",
       "       1.58044106e-06, 1.57544754e-06, 1.57109822e-06, 1.56611012e-06,\n",
       "       1.56607437e-06, 1.55791626e-06, 1.55410906e-06, 1.54659609e-06,\n",
       "       1.54280934e-06, 1.52975471e-06, 1.52799153e-06, 1.52685570e-06,\n",
       "       1.52187915e-06, 1.51961635e-06, 1.51705441e-06, 1.50468559e-06,\n",
       "       1.49616181e-06, 1.48548683e-06, 1.48313252e-06, 1.48126258e-06,\n",
       "       1.47777521e-06, 1.46761264e-06, 1.46504451e-06, 1.46234853e-06,\n",
       "       1.45892850e-06, 1.44808319e-06, 1.44589586e-06, 1.44129570e-06,\n",
       "       1.44011300e-06, 1.43542034e-06, 1.42680821e-06, 1.42635698e-06,\n",
       "       1.42203230e-06, 1.41825049e-06, 1.41051791e-06, 1.40641723e-06,\n",
       "       1.38834066e-06, 1.38310488e-06, 1.37313887e-06, 1.37266076e-06,\n",
       "       1.36596577e-06, 1.35987890e-06, 1.35918149e-06, 1.35416142e-06,\n",
       "       1.34662312e-06, 1.34410881e-06, 1.33589355e-06, 1.33363886e-06,\n",
       "       1.33153972e-06, 1.32739220e-06, 1.31728268e-06, 1.31267121e-06,\n",
       "       1.31252330e-06, 1.29943743e-06, 1.29812489e-06, 1.28847966e-06,\n",
       "       1.28609452e-06, 1.27646105e-06, 1.27552448e-06, 1.27092768e-06,\n",
       "       1.26970916e-06, 1.25799474e-06, 1.25695415e-06, 1.25299550e-06,\n",
       "       1.24214070e-06, 1.24031101e-06, 1.23754506e-06, 1.23298233e-06,\n",
       "       1.22238296e-06, 1.21789301e-06, 1.21416541e-06, 1.21382636e-06,\n",
       "       1.20959369e-06, 1.19567356e-06, 1.19383626e-06, 1.18416358e-06,\n",
       "       1.17908513e-06, 1.17854759e-06, 1.17713341e-06, 1.17221323e-06,\n",
       "       1.16969669e-06, 1.15694348e-06, 1.15246983e-06, 1.15057659e-06,\n",
       "       1.14542583e-06, 1.14416032e-06, 1.14379129e-06, 1.13981272e-06,\n",
       "       1.12506203e-06, 1.12008510e-06, 1.11670997e-06, 1.11106823e-06,\n",
       "       1.10966435e-06, 1.10578644e-06, 1.10072903e-06, 1.09497897e-06,\n",
       "       1.09314930e-06, 1.08830615e-06, 1.08401106e-06, 1.08202703e-06,\n",
       "       1.07568963e-06, 1.06938906e-06, 1.06493675e-06, 1.05703134e-06,\n",
       "       1.05366569e-06, 1.05163487e-06, 1.05027342e-06, 1.03953101e-06,\n",
       "       1.03629895e-06, 1.03392248e-06, 1.02351777e-06, 1.02015570e-06,\n",
       "       1.01615715e-06, 1.00641598e-06, 1.00178061e-06, 9.95992457e-07,\n",
       "       9.92324321e-07, 9.91197789e-07, 9.88179942e-07, 9.76870407e-07,\n",
       "       9.72283325e-07, 9.69993141e-07, 9.65796184e-07, 9.59234422e-07,\n",
       "       9.52664931e-07, 9.49372679e-07, 9.46681202e-07, 9.39255664e-07,\n",
       "       9.34403838e-07, 9.28615166e-07, 9.28356582e-07, 9.19728413e-07,\n",
       "       9.15388996e-07, 9.12558741e-07, 9.09936394e-07, 9.05551680e-07,\n",
       "       8.98795677e-07, 8.93798386e-07, 8.91901087e-07, 8.89882207e-07,\n",
       "       8.82488305e-07, 8.81378125e-07, 8.77814867e-07, 8.77465824e-07,\n",
       "       8.69395142e-07, 8.66280302e-07, 8.56732602e-07, 8.54413949e-07,\n",
       "       8.46602894e-07, 8.45925367e-07, 8.39612851e-07, 8.35541351e-07,\n",
       "       8.32748859e-07, 8.24637929e-07, 8.23678709e-07, 8.16693576e-07,\n",
       "       8.15824376e-07, 8.11936740e-07, 7.98430834e-07, 7.95950672e-07,\n",
       "       7.95112084e-07, 7.88371017e-07, 7.86893468e-07, 7.85988369e-07,\n",
       "       7.80400777e-07, 7.76411085e-07, 7.73502849e-07, 7.68720967e-07,\n",
       "       7.63751914e-07, 7.55493985e-07, 7.54625014e-07, 7.48169683e-07,\n",
       "       7.47410086e-07, 7.43380337e-07, 7.32084887e-07, 7.29451102e-07,\n",
       "       7.29354824e-07, 7.15766864e-07, 7.12019093e-07, 7.10452283e-07,\n",
       "       7.00034053e-07, 6.95061478e-07, 6.92365037e-07, 6.87277886e-07,\n",
       "       6.80976395e-07, 6.76748349e-07, 6.71172898e-07, 6.67471136e-07,\n",
       "       6.65272570e-07, 6.61551991e-07, 6.52071885e-07, 6.43425308e-07,\n",
       "       6.42201165e-07, 6.39661593e-07, 6.34388112e-07, 6.29878999e-07,\n",
       "       6.21621865e-07, 6.21073391e-07, 6.18634099e-07, 6.17796216e-07,\n",
       "       6.14399180e-07, 6.10587689e-07, 6.00286569e-07, 5.98647840e-07,\n",
       "       5.95246855e-07, 5.94221016e-07, 5.84781349e-07, 5.83707916e-07,\n",
       "       5.79264248e-07, 5.78338034e-07, 5.68123858e-07, 5.65928071e-07,\n",
       "       5.64024268e-07, 5.57275474e-07, 5.53016152e-07, 5.46531078e-07,\n",
       "       5.42123603e-07, 5.39233978e-07, 5.34917918e-07, 5.33545784e-07,\n",
       "       5.26142838e-07, 5.23850827e-07, 5.19986098e-07, 5.13854292e-07,\n",
       "       5.08446805e-07, 5.01811050e-07, 4.99770900e-07, 4.93866443e-07,\n",
       "       4.90561839e-07, 4.84279113e-07, 4.82856930e-07, 4.80184377e-07,\n",
       "       4.79422618e-07, 4.77599066e-07, 4.70840560e-07, 4.68841544e-07,\n",
       "       4.65812411e-07, 4.59574450e-07, 4.50965136e-07, 4.45425098e-07,\n",
       "       4.42153515e-07, 4.38003131e-07, 4.32522796e-07, 4.31363780e-07,\n",
       "       4.26385425e-07, 4.23679430e-07, 4.16645785e-07, 4.15482397e-07,\n",
       "       4.09471829e-07, 4.07600053e-07, 4.02520134e-07, 3.99100130e-07,\n",
       "       3.87660431e-07, 3.84372361e-07, 3.81326875e-07, 3.78613071e-07,\n",
       "       3.73508332e-07, 3.73051482e-07, 3.66754769e-07, 3.61691588e-07,\n",
       "       3.58464818e-07, 3.50306600e-07, 3.48276824e-07, 3.38116309e-07,\n",
       "       3.36856609e-07, 3.34548519e-07, 3.32773302e-07, 3.31504132e-07,\n",
       "       3.28780127e-07, 3.14120099e-07, 3.13492573e-07, 3.05066784e-07,\n",
       "       3.04229035e-07, 2.99065641e-07, 2.98814637e-07, 2.93516115e-07,\n",
       "       2.86997086e-07, 2.86426876e-07, 2.76023370e-07, 2.70311775e-07,\n",
       "       2.66013454e-07, 2.65653666e-07, 2.62001490e-07, 2.50823973e-07,\n",
       "       2.45167615e-07, 2.43291142e-07, 2.42662930e-07, 2.32139499e-07,\n",
       "       2.29426431e-07, 2.25648514e-07, 2.22355007e-07, 2.20516920e-07,\n",
       "       2.12374911e-07, 2.07739509e-07, 2.07404862e-07, 2.03178663e-07,\n",
       "       1.99762205e-07, 1.95673929e-07, 1.91135826e-07, 1.86693939e-07,\n",
       "       1.82568205e-07, 1.73510822e-07, 1.70724545e-07, 1.62572357e-07,\n",
       "       1.61688997e-07, 1.58838177e-07, 1.57870423e-07, 1.53433323e-07,\n",
       "       1.50592576e-07, 1.47900560e-07, 1.36496603e-07, 1.33317319e-07,\n",
       "       1.30074048e-07, 1.25140317e-07, 1.21534314e-07, 1.12379957e-07,\n",
       "       1.10496181e-07, 1.04468736e-07, 1.04468020e-07, 9.51792106e-08,\n",
       "       9.51380789e-08, 8.59939258e-08, 8.51781594e-08, 7.47553919e-08,\n",
       "       7.39788601e-08, 7.08508319e-08, 6.83526577e-08, 6.12276536e-08,\n",
       "       6.00709867e-08, 5.45758147e-08, 4.92408389e-08, 4.41025575e-08,\n",
       "       4.34835308e-08, 3.89683177e-08, 3.38756010e-08, 3.24314071e-08,\n",
       "       2.27105727e-08, 1.90492180e-08, 1.21300820e-08, 9.45965781e-09,\n",
       "       8.48541208e-09, 3.74321308e-09, 3.51903489e-09, 2.35622087e-15])"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "fa2f4716",
   "metadata": {},
   "outputs": [],
   "source": [
    "VT = pd.DataFrame(VT)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "7161386b",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <th>490</th>\n",
       "      <th>491</th>\n",
       "      <th>492</th>\n",
       "      <th>493</th>\n",
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       "      <th>499</th>\n",
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       "      <td>0.018255</td>\n",
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       "      <td>-0.014835</td>\n",
       "      <td>-0.015570</td>\n",
       "      <td>-0.003109</td>\n",
       "      <td>0.011545</td>\n",
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       "      <td>-0.011286</td>\n",
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       "      <th>1</th>\n",
       "      <td>0.023537</td>\n",
       "      <td>0.005218</td>\n",
       "      <td>0.006176</td>\n",
       "      <td>0.003314</td>\n",
       "      <td>0.042395</td>\n",
       "      <td>0.036854</td>\n",
       "      <td>0.018510</td>\n",
       "      <td>0.002920</td>\n",
       "      <td>0.009807</td>\n",
       "      <td>-0.021750</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.011303</td>\n",
       "      <td>-0.011063</td>\n",
       "      <td>-0.017403</td>\n",
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       "      <td>-0.035706</td>\n",
       "      <td>-0.017246</td>\n",
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       "      <td>-0.004673</td>\n",
       "      <td>0.014919</td>\n",
       "      <td>0.027448</td>\n",
       "      <td>0.021919</td>\n",
       "      <td>0.023584</td>\n",
       "      <td>0.030860</td>\n",
       "      <td>-0.002765</td>\n",
       "      <td>0.024576</td>\n",
       "      <td>0.020967</td>\n",
       "      <td>0.101282</td>\n",
       "      <td>...</td>\n",
       "      <td>0.079818</td>\n",
       "      <td>0.086883</td>\n",
       "      <td>0.023343</td>\n",
       "      <td>-0.012814</td>\n",
       "      <td>0.033091</td>\n",
       "      <td>0.021553</td>\n",
       "      <td>0.000365</td>\n",
       "      <td>-0.037298</td>\n",
       "      <td>-0.042501</td>\n",
       "      <td>-0.023948</td>\n",
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       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.027282</td>\n",
       "      <td>-0.048530</td>\n",
       "      <td>0.044896</td>\n",
       "      <td>0.007954</td>\n",
       "      <td>0.022463</td>\n",
       "      <td>0.026459</td>\n",
       "      <td>-0.035689</td>\n",
       "      <td>-0.030301</td>\n",
       "      <td>0.013820</td>\n",
       "      <td>0.045942</td>\n",
       "      <td>...</td>\n",
       "      <td>0.024963</td>\n",
       "      <td>0.014741</td>\n",
       "      <td>-0.020335</td>\n",
       "      <td>-0.019253</td>\n",
       "      <td>0.031483</td>\n",
       "      <td>-0.036509</td>\n",
       "      <td>-0.074290</td>\n",
       "      <td>-0.015249</td>\n",
       "      <td>0.032443</td>\n",
       "      <td>0.089253</td>\n",
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       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>-0.048683</td>\n",
       "      <td>-0.016666</td>\n",
       "      <td>-0.030452</td>\n",
       "      <td>-0.000284</td>\n",
       "      <td>-0.041785</td>\n",
       "      <td>-0.024583</td>\n",
       "      <td>0.020820</td>\n",
       "      <td>-0.047374</td>\n",
       "      <td>0.000348</td>\n",
       "      <td>-0.007839</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.018395</td>\n",
       "      <td>-0.045159</td>\n",
       "      <td>0.013354</td>\n",
       "      <td>-0.037162</td>\n",
       "      <td>-0.032157</td>\n",
       "      <td>0.071049</td>\n",
       "      <td>-0.008611</td>\n",
       "      <td>-0.027140</td>\n",
       "      <td>0.023240</td>\n",
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       "      <td>...</td>\n",
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       "      <td>...</td>\n",
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       "      <td>-0.006870</td>\n",
       "      <td>-0.010960</td>\n",
       "      <td>-0.011996</td>\n",
       "      <td>0.043915</td>\n",
       "      <td>-0.055611</td>\n",
       "      <td>0.022623</td>\n",
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       "      <td>-0.054625</td>\n",
       "      <td>-0.031026</td>\n",
       "      <td>0.003804</td>\n",
       "      <td>-0.090840</td>\n",
       "      <td>0.031022</td>\n",
       "      <td>-0.064265</td>\n",
       "      <td>0.085795</td>\n",
       "      <td>0.050347</td>\n",
       "      <td>0.040020</td>\n",
       "      <td>-0.022778</td>\n",
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       "    <tr>\n",
       "      <th>496</th>\n",
       "      <td>-0.027347</td>\n",
       "      <td>-0.067255</td>\n",
       "      <td>0.006137</td>\n",
       "      <td>-0.013406</td>\n",
       "      <td>0.038308</td>\n",
       "      <td>-0.004481</td>\n",
       "      <td>0.039573</td>\n",
       "      <td>0.007133</td>\n",
       "      <td>0.009797</td>\n",
       "      <td>-0.093589</td>\n",
       "      <td>...</td>\n",
       "      <td>0.019867</td>\n",
       "      <td>-0.044339</td>\n",
       "      <td>-0.041987</td>\n",
       "      <td>-0.069452</td>\n",
       "      <td>-0.059265</td>\n",
       "      <td>-0.015115</td>\n",
       "      <td>-0.018858</td>\n",
       "      <td>-0.031000</td>\n",
       "      <td>0.013491</td>\n",
       "      <td>-0.001411</td>\n",
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       "      <th>497</th>\n",
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       "      <td>0.099489</td>\n",
       "      <td>0.034253</td>\n",
       "      <td>-0.023583</td>\n",
       "      <td>0.028885</td>\n",
       "      <td>0.028907</td>\n",
       "      <td>-0.023617</td>\n",
       "      <td>0.042867</td>\n",
       "      <td>-0.032751</td>\n",
       "      <td>...</td>\n",
       "      <td>0.055534</td>\n",
       "      <td>0.009855</td>\n",
       "      <td>-0.038935</td>\n",
       "      <td>0.092464</td>\n",
       "      <td>0.001123</td>\n",
       "      <td>0.079277</td>\n",
       "      <td>0.029559</td>\n",
       "      <td>0.001908</td>\n",
       "      <td>0.037706</td>\n",
       "      <td>-0.057886</td>\n",
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       "    <tr>\n",
       "      <th>498</th>\n",
       "      <td>-0.039187</td>\n",
       "      <td>0.052487</td>\n",
       "      <td>0.025765</td>\n",
       "      <td>0.014198</td>\n",
       "      <td>0.044797</td>\n",
       "      <td>0.025664</td>\n",
       "      <td>-0.032950</td>\n",
       "      <td>0.007805</td>\n",
       "      <td>-0.028169</td>\n",
       "      <td>0.083690</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.028993</td>\n",
       "      <td>0.027184</td>\n",
       "      <td>0.035850</td>\n",
       "      <td>0.011074</td>\n",
       "      <td>0.056673</td>\n",
       "      <td>0.016957</td>\n",
       "      <td>0.017190</td>\n",
       "      <td>-0.065878</td>\n",
       "      <td>-0.037748</td>\n",
       "      <td>-0.003453</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>499</th>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "      <td>-0.044721</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>500 rows × 500 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "          0         1         2         3         4         5         6    \\\n",
       "0   -0.006387 -0.010837 -0.009057 -0.010602 -0.017506 -0.020772  0.002205   \n",
       "1    0.023537  0.005218  0.006176  0.003314  0.042395  0.036854  0.018510   \n",
       "2   -0.004673  0.014919  0.027448  0.021919  0.023584  0.030860 -0.002765   \n",
       "3    0.027282 -0.048530  0.044896  0.007954  0.022463  0.026459 -0.035689   \n",
       "4   -0.048683 -0.016666 -0.030452 -0.000284 -0.041785 -0.024583  0.020820   \n",
       "..        ...       ...       ...       ...       ...       ...       ...   \n",
       "495 -0.011797 -0.070397 -0.008014 -0.041899 -0.006870 -0.010960 -0.011996   \n",
       "496 -0.027347 -0.067255  0.006137 -0.013406  0.038308 -0.004481  0.039573   \n",
       "497 -0.005959  0.004390  0.099489  0.034253 -0.023583  0.028885  0.028907   \n",
       "498 -0.039187  0.052487  0.025765  0.014198  0.044797  0.025664 -0.032950   \n",
       "499 -0.044721 -0.044721 -0.044721 -0.044721 -0.044721 -0.044721 -0.044721   \n",
       "\n",
       "          7         8         9    ...       490       491       492  \\\n",
       "0   -0.007663 -0.006408 -0.021189  ...  0.010054  0.018009  0.018255   \n",
       "1    0.002920  0.009807 -0.021750  ... -0.011303 -0.011063 -0.017403   \n",
       "2    0.024576  0.020967  0.101282  ...  0.079818  0.086883  0.023343   \n",
       "3   -0.030301  0.013820  0.045942  ...  0.024963  0.014741 -0.020335   \n",
       "4   -0.047374  0.000348 -0.007839  ... -0.018395 -0.045159  0.013354   \n",
       "..        ...       ...       ...  ...       ...       ...       ...   \n",
       "495  0.043915 -0.055611  0.022623  ... -0.054625 -0.031026  0.003804   \n",
       "496  0.007133  0.009797 -0.093589  ...  0.019867 -0.044339 -0.041987   \n",
       "497 -0.023617  0.042867 -0.032751  ...  0.055534  0.009855 -0.038935   \n",
       "498  0.007805 -0.028169  0.083690  ... -0.028993  0.027184  0.035850   \n",
       "499 -0.044721 -0.044721 -0.044721  ... -0.044721 -0.044721 -0.044721   \n",
       "\n",
       "          493       494       495       496       497       498       499  \n",
       "0    0.030588 -0.014835 -0.015570 -0.003109  0.011545 -0.002565 -0.011286  \n",
       "1   -0.003624 -0.026346 -0.035488 -0.035706 -0.017246 -0.008398 -0.008489  \n",
       "2   -0.012814  0.033091  0.021553  0.000365 -0.037298 -0.042501 -0.023948  \n",
       "3   -0.019253  0.031483 -0.036509 -0.074290 -0.015249  0.032443  0.089253  \n",
       "4   -0.037162 -0.032157  0.071049 -0.008611 -0.027140  0.023240  0.056195  \n",
       "..        ...       ...       ...       ...       ...       ...       ...  \n",
       "495 -0.090840  0.031022 -0.064265  0.085795  0.050347  0.040020 -0.022778  \n",
       "496 -0.069452 -0.059265 -0.015115 -0.018858 -0.031000  0.013491 -0.001411  \n",
       "497  0.092464  0.001123  0.079277  0.029559  0.001908  0.037706 -0.057886  \n",
       "498  0.011074  0.056673  0.016957  0.017190 -0.065878 -0.037748 -0.003453  \n",
       "499 -0.044721 -0.044721 -0.044721 -0.044721 -0.044721 -0.044721 -0.044721  \n",
       "\n",
       "[500 rows x 500 columns]"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "VT"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "deee0188",
   "metadata": {},
   "source": [
    "reference \n",
    "\n",
    "- scipy, [here](https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.svd.html)\n",
    "- numpy, [here](https://numpy.org/doc/stable/reference/generated/numpy.linalg.svd.html)\n",
    "- matlab, [here](https://www.mathworks.com/help/matlab/ref/double.svd.html#bu2_0hq-U)\n",
    "- other, [here](https://machinelearningmastery.com/singular-value-decomposition-for-machine-learning/)"
   ]
  }
 ],
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