Linear-Algebra-For-ML/matrix-transformations-and-eigen-decomposition.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "945b97d3",
   "metadata": {},
   "source": [
    "# Imports"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "9ac893b5",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-11-20T04:18:55.271560Z",
     "start_time": "2022-11-20T04:18:54.607446Z"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "53b6859d",
   "metadata": {},
   "source": [
    "# Visualizing vectors in 2D\n",
    "\n",
    "The `plot_vectors` function plots a list of vectors in 2D space."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "cc7e895c",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-11-20T04:18:55.302811Z",
     "start_time": "2022-11-20T04:18:55.272228Z"
    }
   },
   "outputs": [],
   "source": [
    "def plot_vectors(vectors, colors, scale=60):\n",
    "    V = np.array([[0, 1], [1, 0]])  # ihat and jhat\n",
    "    colors = ['k', 'k'] + colors\n",
    "    V = np.vstack([V, np.hstack(vectors).T])\n",
    "    \n",
    "    plt.figure(figsize=(5, 5))\n",
    "    plt.grid()\n",
    "    \n",
    "    origin = np.zeros(V.T.shape)\n",
    "    plt.quiver(*origin, V[:,0], V[:,1], color=colors, scale=1, units='xy')\n",
    "    \n",
    "    lim = V.max() * 1.5\n",
    "    \n",
    "    plt.xlim(-lim, lim)\n",
    "    plt.ylim(-lim, lim)\n",
    "    \n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "abcd5b03",
   "metadata": {},
   "source": [
    "Let's look at some examples - $v_1$ and $v_2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "4640cc16",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-11-20T04:18:55.318741Z",
     "start_time": "2022-11-20T04:18:55.303812Z"
    }
   },
   "outputs": [],
   "source": [
    "v1 = np.array([[2, 2]]).T\n",
    "v2 = np.array([[2, 1]]).T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "a42e6e8b",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-11-20T04:18:55.413089Z",
     "start_time": "2022-11-20T04:18:55.320752Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_vectors([v1, v2], colors=['r', 'g'])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7ddc3e39",
   "metadata": {},
   "source": [
    "# Matrix Transformation\n",
    "\n",
    "Say, $M$ is a 2D transformation matrix. Let's apply $M$ on some random 2D vectors and see where they land after being transformed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "45a98098",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-11-20T04:18:55.428886Z",
     "start_time": "2022-11-20T04:18:55.414089Z"
    }
   },
   "outputs": [],
   "source": [
    "# M = np.array([\n",
    "#     [3, 1],\n",
    "#     [0, 2]\n",
    "# ])\n",
    "\n",
    "M = np.array([\n",
    "    [-0.5, 1],\n",
    "    [1, 0]\n",
    "])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b5bf0330",
   "metadata": {},
   "source": [
    "Say, $v = \\hat{i} + 2\\hat{j}$ and the transformed vector is $v_m = Mv$\n",
    "\n",
    "<!-- #### Note: L2 Norm\n",
    "<img src=\"L2.png\" width=\"450\" height=\"300\"> -->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "2394f760",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-11-20T04:18:55.508957Z",
     "start_time": "2022-11-20T04:18:55.429912Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# v = np.array([[1, 2]]).T\n",
    "# v_m = M.dot(v)\n",
    "# print(f'Length scaled = {np.linalg.norm(v_m) / np.linalg.norm(v)}')\n",
    "# plot_vectors([v, v_m], colors=['r', 'g'])\n",
    "\n",
    "v1 = np.array([[2, 2]]).T\n",
    "v2 = np.array([[-2, 3]]).T\n",
    "\n",
    "v1_m = M.dot(v1)\n",
    "v2_m = M.dot(v2)\n",
    "\n",
    "plot_vectors([v1, v2, v1_m, v2_m], colors=['purple', 'purple', 'orange', 'orange'])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "24d8ce50",
   "metadata": {},
   "source": [
    "#### Old Notebook Version\n",
    "\n",
    "Red arrow shows the original vector and the green arrow shows the transformed vector. \n",
    "\n",
    "After transformation, two things happen to the transformed vector\n",
    "\n",
    "1. It may change direction\n",
    "2. It may change length"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e37a4766",
   "metadata": {},
   "source": [
    "Let's see how our basis vectors $\\hat{i}$ and $\\hat{j}$ gets transformed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "e48a1d0d",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-11-20T04:18:55.603140Z",
     "start_time": "2022-11-20T04:18:55.509925Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Transformed vector: [[-0.5  1. ]]\n",
      "1st column of M: [-0.5  1. ]\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "v = np.array([[1, 0]]).T # i_hat\n",
    "v_m = M.dot(v)\n",
    "print(f'Transformed vector: {v_m.T}')\n",
    "print(f'1st column of M: {M[:, 0]}')\n",
    "plot_vectors([v, v_m], colors=['r', 'g'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "de7a819a",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-11-20T04:18:55.682352Z",
     "start_time": "2022-11-20T04:18:55.604151Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Transformed vector: [[1. 0.]]\n",
      "2nd column of M: [1. 0.]\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "v = np.array([[0, 1]]).T  # j_hat\n",
    "v_m = M.dot(v)\n",
    "print(f'Transformed vector: {v_m.T}')\n",
    "print(f'2nd column of M: {M[:, 1]}')\n",
    "plot_vectors([v, v_m], colors=['r', 'g'])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c914445a",
   "metadata": {},
   "source": [
    "Can you notice the similarities between transformed vectors and colums of $M$? [Hint](https://i.imgur.com/mWQjR2C.png)\n",
    "\n",
    "Please watch [the amazing explanation](https://youtu.be/kYB8IZa5AuE) from [3Blue1Brown](https://www.youtube.com/@3blue1brown)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "55daddd0",
   "metadata": {},
   "source": [
    "# Task 1\n",
    "\n",
    "Steps for determining the transformation matrix $M$ for a desired transformation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "1ac637a0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "v1 = [[2 2]]\n",
      "v2 = [[-2  3]]\n"
     ]
    }
   ],
   "source": [
    "# Given vectors\n",
    "v1 = np.array([[2, 2]]).T\n",
    "v2 = np.array([[-2, 3]]).T\n",
    "\n",
    "# Print the vectors\n",
    "print(f'v1 = {v1.T}')\n",
    "print(f'v2 = {v2.T}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "4f01aa6d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot before transformation\n",
    "plot_vectors([v1, v2], colors=['purple', 'purple'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "5aecb1b5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Matrix m:\n",
      "[[ 2 -2]\n",
      " [ 2  3]]\n"
     ]
    }
   ],
   "source": [
    "# Create a matrix combining the two vectors\n",
    "m = np.hstack([v1, v2])\n",
    "print(f'Matrix m:\\n{m}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "50bbf0ca",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Inverse of m:\n",
      "[[ 0.3  0.2]\n",
      " [-0.2  0.2]]\n"
     ]
    }
   ],
   "source": [
    "# Inverse the matrix\n",
    "m_inv = np.linalg.inv(m)\n",
    "print(f'Inverse of m:\\n{m_inv}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "cf199776",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "v3 = [[ 4 -2]]\n",
      "v4 = [[1 2]]\n"
     ]
    }
   ],
   "source": [
    "# Target vector after transformation\n",
    "v3 = np.array([[4, -2]]).T\n",
    "v4 = np.array([[1, 2]]).T\n",
    "\n",
    "# Print the vectors\n",
    "print(f'v3 = {v3.T}')\n",
    "print(f'v4 = {v4.T}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "007bee9c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 500x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot after transformation\n",
    "plot_vectors([v3, v4], colors=['orange', 'orange'])\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "25c3a16a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Matrix m_new:\n",
      "[[ 1  4]\n",
      " [ 2 -2]]\n"
     ]
    }
   ],
   "source": [
    "# Create a matrix combining the two vectors\n",
    "\n",
    "m_new = np.hstack([v4, v3])\n",
    "print(f'Matrix m_new:\\n{m_new}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "36380881",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Transformation matrix:\n",
      "[[-0.5  1. ]\n",
      " [ 1.   0. ]]\n"
     ]
    }
   ],
   "source": [
    "# Find out the transformation matrix\n",
    "\n",
    "M = np.dot(m_new, m_inv)\n",
    "print(f'Transformation matrix:\\n{M}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7eee262a",
   "metadata": {},
   "source": [
    "# Eigen Decomposition"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5b0b11ee",
   "metadata": {},
   "source": [
    "Let's look at a special vector, $v_e = \\hat{i}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "263736bf",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-11-20T04:18:55.778003Z",
     "start_time": "2022-11-20T04:18:55.683322Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Length scaled 1.118033988749895\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "v_e = np.array([[1, 0]]).T\n",
    "v_em = M.dot(v_e)\n",
    "print(f'Length scaled {np.linalg.norm(v_em) / np.linalg.norm(v_e)}')\n",
    "plot_vectors([v_e, v_em], colors=['r', 'g'])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b57bf2a6",
   "metadata": {},
   "source": [
    "As you can see the transformed vector stayed on the same line as $v_e$.\n",
    "\n",
    "These kinds of vectors are called eigenvectors of $M$.\n",
    "\n",
    "You can find these by applying eigenvalue decomposition."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "507f4e38",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-11-20T04:18:55.794110Z",
     "start_time": "2022-11-20T04:18:55.779995Z"
    }
   },
   "outputs": [],
   "source": [
    "eigen_values, eigen_vectors = np.linalg.eig(M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "6a116ede",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-11-20T04:18:55.810174Z",
     "start_time": "2022-11-20T04:18:55.795110Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1.28077641,  0.78077641])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigen_values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "1942351b",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-11-20T04:18:55.825477Z",
     "start_time": "2022-11-20T04:18:55.811148Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.78820544, -0.61541221],\n",
       "       [ 0.61541221, -0.78820544]])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigen_vectors"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cd862bf3",
   "metadata": {},
   "source": [
    "Let's plot these before and after transformation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "32bd2c38",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2022-11-20T04:18:55.999437Z",
     "start_time": "2022-11-20T04:18:55.826476Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenvector [[-0.78820544  0.61541221]]\n",
      "Eigenvalue -1.2807764064044151\n",
      "Length scaled 1.2807764064044151\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 500x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenvector [[-0.61541221 -0.78820544]]\n",
      "Eigenvalue 0.7807764064044151\n",
      "Length scaled 0.7807764064044151\n"
     ]
    },
    {
     "data": {
      "image/png": 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hlT63Z88ezZ8/X/Pnz1dkZKQGDBjgPVDxx9fASXaM3dbQ/Omnn1RcXKyoqKgy/VFRUfryyy8rXCcrK6vC5bOysipcPiUlRTNmzCjXv3btWoWFhVWz8vohNTXV6RIc549z0L/vDPX/5efFixc7Wosv8Nc5yMnJkeSfr4GTjh49WuPbtP30rN2mTp1a5sg0Pz9fLVq0UN++fRUZGelgZc7xeDxKTU1V//79FRQU5HQ5jvDHOfh6S6oS3rlGBcFSaECoFndYrHPOOUcul8vp0hxhWZYOHz5cL+bgzTff1JNPPlmu/6yzztLll1+ugQMHasCAAWrWrJn3OX98Dfy3k/9wqEm2hmaTJk0UGBio7OzsMv3Z2dmKjo6ucJ3o6OgqLe92u+V2u8v1BwUF+e2OchJz4D9zUHAoS8OXXqUfGxVKJVK3vLMkSfHx8X4x/op4PB6tXLmyzs9BUVGR/vSnP3lPMbdo0UJDhgzRkCFDlJCQoJCQkFOu7y+vgYrYMW5br54NDg5W9+7dlZaW5u0rKSlRWlqa4uPjK1wnPj6+zPJS6emFypYH/J1VUqJbHuipLxqVvrff5JhLL4xe4WxRqDGvvPKKIiMj9cADD+g///mP9u3bpwULFmjgwIGnDUzUPNtPzyYnJ2v06NHq0aOHevbsqXnz5unIkSPeN6lHjRqlc889VykpKZKkiRMnqk+fPnrkkUd0xRVX6JVXXtGmTZv09NNP210qUCfNn3ONljb6VpIUUCK98j+z1Tyup7bt4f7m+mD48OG68cYbnS4Dv7A9NK+77jr9+OOPmjZtmrKystSlSxetWrXKe7HP/v37FRDw6wFv79699dJLL+m+++7Tvffeq/PPP18rVqxQhw4d7C4VqHM+fvsJ3VWwXPrlDoMHggao39V3+/UVk/VNRW8/wTm1ciHQhAkTNGHChAqfW7duXbm+4cOHa/jw4TZXBdRtB3Zn6tqPJ6qo9O1LDc2N1j1z3nW2KKCe47NngTrIc/yorn28j7LOKpEktcsP0nNTMxQQWOcviAd8GqEJ1EGTZ1yqT84p/SCPsBPSG1e+rEbNWjpcFVD/EZpAHfPK03/TvJCt3vailhPU8dKrHawI8B+EJlCHbN/wpsbue9zb/tvxzhr558dPsQaAmkRoAnVE/o/f6arXh+tocGn7d4fP1sPTPnG2KMDPEJpAHWCVlOjm/+2pr8JLbyWJOhqgV29fp+DQhg5XBvgXQhOoAx5O+YOWRxyQJAWWSK/+bp5iz+/mcFWA/yE0AR+35o1HNPXEe972Q6F/1GV//KuDFQH+i9AEfNi3OzI0IuNulfzySh2e11x3TF7ubFGAHyM0AR9VeCRfw5/upx/DLEnShXnBeubvGXIF8LIFnMKrD/BRd8zsrYyII5KkhiekN0Ys19mRsQ5XBfg3QhPwQc8tGK//C9vubS9pe5fieg52sCIAEqEJ+JzMD5fq1gP/z9u++8T/6Oo/PexgRQBO4tOdAR9y+MBeXf32jTp+dmk74XCE/vehj5wtCoAXR5qAjygpLtJND12iPWcXSZLOLQjQKxM/VoPgEIcrA3ASoQn4iAcfTNK7EQclSUHF0rLL/09RrfnydcCXEJqAD1j1ygOaXrLG254bPlzxg8Y7WBGAihCagMP2bvtI1/9nmixXafvG/Nb6S/IrzhYFoEKEJuCgY/mHdM2zA3U4pPQDDDrlhuipaRv5AAPAR/HKBBxiWZZun3WJtkQckyQ1Oi69PuodhTVq4nBlACpDaAIOWTR/tJ5t+LW3/cJF96ld134OVgTgdAhNwAH/Tn1OE356wdu+r+RSDblploMVATBBaAK17Kdvd+qaD8bqxC8fLTLgcKTuvy/N2aIAGCE0gVpU7Dmh6x/prf0NiyVJrX4O1Et3pSswKNjhygCYIDSBWjR91uVKPeeQJCm4SHpt4GJFNj/f4aoAmCI0gVry1gt/14OB673tBU1GqUfiKAcrAlBVhCZQC77enKqbdvyvtz224ALdMvE5BysCUB2EJmCzI4cP6uoX/6h8d2m7e26Ynpie4WxRAKqF0ARsZJWUaPwDvfRZo+OSpMbHXHpt7PsKaRjhbGEAqoXQBGy04NERein8G0mSy5Je6jpL53W41NmiAFQboQnYZMPKp3RH/jJve2ZAPyVd93cHKwJwpghNwAbZez/X8HV/UVFgafsPuc10799XOVsUgDNGaAI1rOjEcV332KX64awSSVKbnxvohXsyFBDYwOHKAJwpQhOoYVNn/F4fnpMnSQrxSG/88UVFRJ/nbFEAagShCdSg1565U3OCN3nbT8WMU+fLrnWwIgA1idAEasiOjHc0Zs+j3vZfjnbQqNufdrAiADWN0ARqwM85P+iqpVer4JfPXb8kt6Hm3p/ubFEAahyhCZwhq6REf3qwl75sdEKS1PSoS8tuXaPg0IYOVwagphGawBma+9CVeq3Rd5KkgBJp6SVz1Lz9/zhcFQA7EJrAGfjwzcc0+dhb3vY/gwer75XJDlYEwE6EJlBN33+1SdduuEPFv7yKrsqN1V1T33a2KAC2IjSBajhxrEDXPnm5DoZZkqT2eUF69t4MuQJ4SQH1Ga9woBrumvE7bTjnZ0nSWSek169ZpvCmzR2uCoDdCE2gil5c+Bc9HrrN237mvL/p4t5DHawIQG0hNIEq2PbJ6xr33f9523cUdtN14x5zsCIAtYnQBAzlZu/T1ctH6lhQafv3h8M1e9rHzhYFoFYRmoCBkuIijf7nJdoV7pEkxRwJ0Kt/+1hBIWEOVwagNhGagIHZKVforYgsSVKDYunVyx5XdJtODlcFoLYRmsBppC77p+7zfOBtP9LwSl36h784WBEApxCawCns/yJdIzffq5JfXikj81rqr3e95mxRABxDaAKVOF6Qq6v/X6JyQks/wODiPLeevo8PMAD8Ga9+oBITZ8ZrU8RRSdLZhdIb17+pho2jHa4KgJMITaACix//k54+60tv+7kLJuuCHkkOVgTAF9gamocOHdINN9yg8PBwRUREaOzYsSooKDjlOgkJCXK5XGUet956q51lAmVsWfOi/nLwWW97StEluvLm2Q5WBMBXNLBz4zfccIMOHDig1NRUeTwejRkzRuPHj9dLL710yvXGjRunmTNnetthYdwLh9px6IfdunrlzSo8u7Td7/A5mvXQWmeLAuAzbAvNHTt2aNWqVfr3v/+tHj16SJIef/xxDR48WHPmzFFsbGyl64aFhSk6mveOULtKiot0w8OX6JuIIklS84JAvZy8Xg2CQxyuDICvsC0009PTFRER4Q1MSUpMTFRAQIAyMjJ05ZVXVrruiy++qH/961+Kjo7WkCFD9I9//KPSo83CwkIVFhZ62/n5+ZIkj8cjj8dTQ6OpW06O21/HL1VvDlJSrtCHjY8oVKEKLpKW9VugiJh2dXIe2QeYA38fv2TP2G0LzaysLDVr1qzsL2vQQI0bN1ZWVlal611//fVq1aqVYmNjtW3bNt1zzz3auXOn3njjjQqXT0lJ0YwZM8r1r1271u9P66ampjpdguOqMgedO9+ul3W7t51dKK1cudKOsmoN+wBz4M/jP3r0aI1vs8qhOWXKFM2efeqLInbs2FHtgsaPH+/9uWPHjoqJiVG/fv20e/dutW3bttzyU6dOVXJysredn5+vFi1aqG/fvoqMjKx2HXWZx+NRamqq+vfvr6CgIKfLccTp5mBe+jzd2PlGNQlrom8+/0SXvf4H5blL78e8vqC1npy5pU7fj8k+wBz4+/glKScnp8a3WeXQvPPOO3XzzTefcpk2bdooOjpaBw8eLNNfVFSkQ4cOVen9yl69ekmSdu3aVWFout1uud3ucv1BQUF+u6OcxBxUPAfFJcWav3m+8ovyNbX7RA1/frCyIo5LJVKX3FA9PuNjBVewT9VF7APMgT+P345xVzk0mzZtqqZNm552ufj4eOXm5mrz5s3q3r27JGnNmjUqKSnxBqGJzMxMSVJMTExVSwUq9MHuD/Rd/nd6fOPj+vq9f+k/EcclSRHHXXr95pUKDW/scIUAfJVt558uvPBCDRw4UOPGjdPGjRu1fv16TZgwQSNGjPBeOfv9998rLi5OGzdulCTt3r1bs2bN0ubNm/XNN9/orbfe0qhRo3TZZZepUye+UQI1Y9HWRZKkvMI8LW34jbf/xY7T1aZzgjNFAagTbH3T5sUXX1RcXJz69eunwYMH69JLL9XTTz/tfd7j8Wjnzp3eN2uDg4O1evVqDRgwQHFxcbrzzjt19dVX6+2337azTPiR7IJsvbXzrXL9v7NaKPbyofIU+++VhgBOz9YPN2jcuPEpP8jgvPPOk2VZ3naLFi304Ycf2lkS/NwL215QUUlRuf71rm/V9amuij07Vu/f+L46NOvgQHUAfF3dvTwQqCLLsrRoy6JKn+/Xup823rKRwARQKVuPNAFfsv7b9dqZs7Ncf3BgsFL6pWjSJZMU4OLfkQAqR2jCbzyzYlq5vouaXqSXrnpJnaM7O1ARgLqGf1bDL+R9vlmvHiz7wet/7flXbRq3icAEYIwjTdR/BQV65d4hOlp6u7CijjfQsze9rkEX/dHZugDUOYQm6jfLksaN0zPRByRJQ/YEadG9n6pZ+24OFwagLuL0LOq3p57Sf9a+os+bSQvfcenNG98lMAFUG0eaqN/+/ndltpG2PCXFJT8o9e/vdEUA6jBCE/VTdnbpf4uKNOo/kuuPf5TuucfZmgDUeZyeRf3j8Ui/+SYeV7t20nPPSXX4q74A+Ab+L4L65557pA0bSn8ODZXeeEOKiHC0JAD1A6GJ+mXpUmnu3F/b8+dLHTs6Vw+AeoXQRP2xfbs0dmzZvmuvdaYWAPUSoYn6IT9fuuoq6ciR0nYVvugcAEwRmqj7LEsaM0b66qvSdlSU9PzzztYEoF4iNFH3zZlTerGPJAUGlr6vGR3tbE0A6iVCE3XPsmW//rxmjTRlyq/t2bOlPn1qvyYAfoHQRN3y+eelp2IPHZK++04aMUIqKSl97pprpORkZ+sDUK/xiUCoW555pvRinzlzSo8yf/yxtD8uTlq8WHK5nK0PQL1GaKLuKCz89QKflJRf+xs2LH1P8+yznakLgN/g9CzqjhUrSk/L/rdHH5UuvLDWywHgfwhN1B2LFlXcP3681LatdPvtpUejAGATQhN1w9690urVlT9//vnSffdJbnft1QTA7xCaqBuefbbifrdbeuwxaeVKKSamdmsC4He4EAi+r7i44tDs1El68UWpQ4farwmAX+JIE77vgw9K78n8rTvukDIyCEwAtYojTfi+314AFBNT+oXS/fs7Vw8Av8WRJnxbdrb01lulP195pbRtG4EJwDGEJnzbCy9IwcGlR5uvvy41aeJ0RQD8GKdn4bssq/TIMjOz9JYSAHAYoQnfVVRU+lmzQUFOVwIAkghN+DLCEoCP4T1NAAAMEZoAABgiNAEAMERoAgBgiNAEAMAQoQkAgCFCEwAAQ4QmAACGCE0AAAwRmgAAGCI0AQAwRGgCAGCI0AQAwBChCQCAIUITAABDhCYAAIYITQAADBGaAAAYIjQBADBEaAIAYIjQBADAkG2h+eCDD6p3794KCwtTRESE0TqWZWnatGmKiYlRaGioEhMT9fXXX9tVIgAAVWJbaJ44cULDhw/XbbfdZrzOQw89pPnz52vhwoXKyMjQWWedpaSkJB0/ftyuMgEAMNbArg3PmDFDkrRkyRKj5S3L0rx583Tfffdp6NChkqTnn39eUVFRWrFihUaMGGFXqQAAGLEtNKtq7969ysrKUmJiorevUaNG6tWrl9LT0ysNzcLCQhUWFnrb+fn5kiSPxyOPx2Nv0T7q5Lj9dfwSc+Dv45eYA38fv2TP2H0mNLOysiRJUVFRZfqjoqK8z1UkJSXFe1T7W2vXrlVYWFjNFlnHpKamOl2C4/x9Dvx9/BJz4M/jP3r0aI1vs0qhOWXKFM2ePfuUy+zYsUNxcXFnVFRVTJ06VcnJyd52fn6+WrRoob59+yoyMrLW6vAlHo9Hqamp6t+/v4KCgpwuxxH+Pgf+Pn6JOfD38UtSTk5OjW+zSqF555136uabbz7lMm3atKlWIdHR0ZKk7OxsxcTEePuzs7PVpUuXStdzu91yu93l+oOCgvx2RzmJOWAO/H38EnPgz+O3Y9xVCs2mTZuqadOmNV6EJLVu3VrR0dFKS0vzhmR+fr4yMjKqdAUuAAB2se2Wk/379yszM1P79+9XcXGxMjMzlZmZqYKCAu8ycXFxWr58uSTJ5XJp0qRJeuCBB/TWW2/ps88+06hRoxQbG6thw4bZVSYAAMZsuxBo2rRpeu6557ztrl27Siq9QCchIUGStHPnTuXl5XmXmTx5so4cOaLx48crNzdXl156qVatWqWQkBC7ygQAwJhtoblkyZLT3qNpWVaZtsvl0syZMzVz5ky7ygIAoNr47FkAAAwRmgAAGCI0AQAwRGgCAGCI0AQAwBChCQCAIUITAABDhCYAAIYITQAADBGaAAAYIjQBADBEaAIAYIjQBADAEKEJAIAhQhMAAEOEJgAAhghNAAAMEZoAABgiNAEAMERoAgBgiNAEAMAQoQkAgCFCEwAAQ4QmAACGCE0AAAwRmgAAGCI0AQAwRGgCAGCI0AQAwBChCQCAIUITAABDhCYAAIYITQAADBGaAAAYIjQBADBEaAIAYIjQBADAEKEJAIAhQhMAAEOEJgAAhghNAAAMEZoAABgiNAEAMERoAgBgiNAEAMAQoQkAgCFCEwAAQ4QmAACGCE0AAAwRmgAAGCI0AQAwRGgCAGCI0AQAwJBtofnggw+qd+/eCgsLU0REhNE6N998s1wuV5nHwIED7SoRAIAqaWDXhk+cOKHhw4crPj5ezzzzjPF6AwcO1LPPPuttu91uO8oDAKDKbAvNGTNmSJKWLFlSpfXcbreio6NtqAgAgDNjW2hW17p169SsWTOdc845uvzyy/XAAw8oMjKy0uULCwtVWFjobefl5UmSDh06ZHutvsrj8ejo0aPKyclRUFCQ0+U4wt/nwN/HLzEH/j5+6dccsCyrxrbpU6E5cOBAXXXVVWrdurV2796te++9V4MGDVJ6eroCAwMrXCclJcV7VPtbF1xwgd3lAgDqgJycHDVq1KhGtuWyqhDBU6ZM0ezZs0+5zI4dOxQXF+dtL1myRJMmTVJubm6Vi9uzZ4/atm2r1atXq1+/fhUu899Hmrm5uWrVqpX2799fY5NU1+Tn56tFixb69ttvFR4e7nQ5jvD3OfD38UvMgb+PXyo989iyZUsdPnzY+ILU06nSkeadd96pm2+++ZTLtGnT5kzqKbetJk2aaNeuXZWGptvtrvBioUaNGvntjnJSeHg4c+Dnc+Dv45eYA38fvyQFBNTcjSJVCs2mTZuqadOmNfbLT+e7775TTk6OYmJiau13AgBQGdvu09y/f78yMzO1f/9+FRcXKzMzU5mZmSooKPAuExcXp+XLl0uSCgoKdPfdd+vTTz/VN998o7S0NA0dOlTt2rVTUlKSXWUCAGDMtguBpk2bpueee87b7tq1qyRp7dq1SkhIkCTt3LnTe7VrYGCgtm3bpueee065ubmKjY3VgAEDNGvWrCrdq+l2uzV9+nS/vr+TOWAO/H38EnPg7+OX7JmDKl0IBACAP+OzZwEAMERoAgBgiNAEAMAQoQkAgKF6EZp8DVn15sCyLE2bNk0xMTEKDQ1VYmKivv76a3sLtcmhQ4d0ww03KDw8XBERERo7dmyZ25sqkpCQUG4fuPXWW2up4jO3YMECnXfeeQoJCVGvXr20cePGUy6/bNkyxcXFKSQkRB07dtTKlStrqVL7VGUOlixZUu7vHRISUovV1qyPPvpIQ4YMUWxsrFwul1asWHHaddatW6du3brJ7XarXbt2Vf5CDV9S1fGvW7eu3N/f5XIpKyurSr+3XoTmya8hu+2226q03sCBA3XgwAHv4+WXX7apQvtVZw4eeughzZ8/XwsXLlRGRobOOussJSUl6fjx4zZWao8bbrhB27dvV2pqqt555x199NFHGj9+/GnXGzduXJl94KGHHqqFas/c0qVLlZycrOnTp2vLli3q3LmzkpKSdPDgwQqX37Bhg0aOHKmxY8dq69atGjZsmIYNG6bPP/+8liuvOVWdA6n003F++/fet29fLVZcs44cOaLOnTtrwYIFRsvv3btXV1xxhfr27avMzExNmjRJt9xyi95//32bK7VHVcd/0s6dO8vsA82aNavaL7bqkWeffdZq1KiR0bKjR4+2hg4dams9TjCdg5KSEis6Otp6+OGHvX25ubmW2+22Xn75ZRsrrHlffPGFJcn697//7e177733LJfLZX3//feVrtenTx9r4sSJtVBhzevZs6d1++23e9vFxcVWbGyslZKSUuHy1157rXXFFVeU6evVq5f15z//2dY67VTVOajK/x/qGknW8uXLT7nM5MmTrYsvvrhM33XXXWclJSXZWFntMBn/2rVrLUnW4cOHz+h31Ysjzeo6+TVk7du312233aacnBynS6o1e/fuVVZWlhITE719jRo1Uq9evZSenu5gZVWXnp6uiIgI9ejRw9uXmJiogIAAZWRknHLdF198UU2aNFGHDh00depUHT161O5yz9iJEye0efPmMn+7gIAAJSYmVvq3S09PL7O8JCUlJdW5v/VJ1ZkDqfSTx1q1aqUWLVpo6NCh2r59e22U6xPq2z5QXV26dFFMTIz69++v9evXV3l9n/pqsNpUna8hq09OnsePiooq0x8VFVXlc/xOy8rKKneKpUGDBmrcuPEpx3L99derVatWio2N1bZt23TPPfdo586deuONN+wu+Yz89NNPKi4urvBv9+WXX1a4TlZWVr34W59UnTlo3769Fi9erE6dOikvL09z5sxR7969tX37djVv3rw2ynZUZftAfn6+jh07ptDQUIcqqx0xMTFauHChevToocLCQi1atEgJCQnKyMhQt27djLfjs6FZna8hq4oRI0Z4f+7YsaM6deqktm3bat26dZV+o0pts3sOfJ3p+Kvrt+95duzYUTExMerXr592796ttm3bVnu78E3x8fGKj4/3tnv37q0LL7xQTz31lGbNmuVgZagN7du3V/v27b3t3r17a/fu3Zo7d65eeOEF4+34bGj64teQ1TY75yA6OlqSlJ2dXeZbZLKzs9WlS5dqbbOmmY4/Ojq63MUfRUVFOnTokHecJnr16iVJ2rVrl0+HZpMmTRQYGKjs7Owy/dnZ2ZWONzo6ukrL+7rqzMF/CwoKUteuXbVr1y47SvQ5le0D4eHh9f4oszI9e/bUJ598UqV1fDY0+Roye+egdevWio6OVlpamjck8/PzlZGRUeWrkO1iOv74+Hjl5uZq8+bN6t69uyRpzZo1Kikp8QahiczMTEnyqX2gIsHBwerevbvS0tI0bNgwSVJJSYnS0tI0YcKECteJj49XWlqaJk2a5O1LTU0tc+RVl1RnDv5bcXGxPvvsMw0ePNjGSn1HfHx8uduM6vI+UBMyMzOr/no/o8uIfMS+ffusrVu3WjNmzLAaNmxobd261dq6dav1888/e5dp37699cYbb1iWZVk///yzddddd1np6enW3r17rdWrV1vdunWzzj//fOv48eNODeOMVHUOLMuy/vnPf1oRERHWm2++aW3bts0aOnSo1bp1a+vYsWNODOGMDBw40OratauVkZFhffLJJ9b5559vjRw50vv8d999Z7Vv397KyMiwLMuydu3aZc2cOdPatGmTtXfvXuvNN9+02rRpY1122WVODaFKXnnlFcvtdltLliyxvvjiC2v8+PFWRESElZWVZVmWZd10003WlClTvMuvX7/eatCggTVnzhxrx44d1vTp062goCDrs88+c2oIZ6yqczBjxgzr/ffft3bv3m1t3rzZGjFihBUSEmJt377dqSGckZ9//tn7OpdkPfroo9bWrVutffv2WZZlWVOmTLFuuukm7/J79uyxwsLCrLvvvtvasWOHtWDBAiswMNBatWqVU0M4I1Ud/9y5c60VK1ZYX3/9tfXZZ59ZEydOtAICAqzVq1dX6ffWi9AcPXq0JancY+3atd5lJFnPPvusZVmWdfToUWvAgAFW06ZNraCgIKtVq1bWuHHjvC+2uqiqc2BZpbed/OMf/7CioqIst9tt9evXz9q5c2ftF18DcnJyrJEjR1oNGza0wsPDrTFjxpT5B8PevXvLzMf+/futyy67zGrcuLHldrutdu3aWXfffbeVl5fn0Aiq7vHHH7datmxpBQcHWz179rQ+/fRT73N9+vSxRo8eXWb5V1991brgggus4OBg6+KLL7befffdWq645lVlDiZNmuRdNioqyho8eLC1ZcsWB6quGSdvofjvx8kxjx492urTp0+5dbp06WIFBwdbbdq0KfP/g7qmquOfPXu21bZtWyskJMRq3LixlZCQYK1Zs6bKv5evBgMAwJBf36cJAEBVEJoAABgiNAEAMERoAgBgiNAEAMAQoQkAgCFCEwAAQ4QmAACGCE0AAAwRmgAAGCI0AQAwRGgCAGDo/wO6oIFIYsBqvgAAAABJRU5ErkJggg==",
      "text/plain": [
       "<Figure size 500x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for i in range(eigen_vectors.shape[0]):\n",
    "    v = eigen_vectors[:, i].reshape(2, 1)\n",
    "    v_m = M.dot(v)\n",
    "    print(f'Eigenvector {v.T}')\n",
    "    print(f'Eigenvalue {eigen_values[i]}')\n",
    "    print(f'Length scaled {np.linalg.norm(v_m) / np.linalg.norm(v)}')\n",
    "    plot_vectors([v, v_m], colors=['r', 'g'])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "80de7157",
   "metadata": {},
   "source": [
    "As you have probably noticed, eigenvalue denotes the amount of scaling after transformation. \n",
    "\n",
    "Let's denote this as $\\lambda$. Then we get a concise equation\n",
    "\n",
    "$$ Mv = \\lambda v $$"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
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