Eigenv_intuition.ipynb

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      "name": "Eigenv_intuition.ipynb",
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      "authorship_tag": "ABX9TyMDbIqTU7cL/XW7GylIpNHf",
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      "name": "python3",
      "display_name": "Python 3"
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  "cells": [
    {
      "cell_type": "markdown",
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        "id": "view-in-github",
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      "source": [
        "<a href=\"https://colab.research.google.com/github/wenxuan0923/My-notes/blob/master/Eigenv_intuition.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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      "cell_type": "markdown",
      "metadata": {
        "id": "b1g3aPKMqD84",
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      "source": [
        "# Intuition behind Eigenvalue, Eigenvector and Eigendecomposition\n",
        "\n",
        "In this note I will explain the intuition behind **eigenvalue**, **eigenvector**, and **eigendecomposition** using visualization, and show you how to calculate them using Numpy. These concepts are very important for us to understand methods like Singular Value Decomposition, Latent Matrix Factorization, Principal Component Analysis and a lot more."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "rBMS9lVu4aZ5",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "import numpy as np\n",
        "import matplotlib\n",
        "import matplotlib.pyplot as plt\n",
        "matplotlib.style.use('seaborn-whitegrid')"
      ],
      "execution_count": 2,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "i6byyfxauGFY",
        "colab_type": "text"
      },
      "source": [
        "First thing first, we need to understand **matrix multiplication** defined as:\n",
        "\n",
        "<br>$$\\mathbf{C_{(m*n)}} = A_{(m*p)}X_{(p*n)}$$ <br>\n",
        "\n",
        "Specificlly, the item on the i-th row and j-th column in $\\mathbf{C}$ is defined as:\n",
        "\n",
        "<br>$$\\mathbf{C_{i,j}} = \\sum_{k=1}^{p}A_{i,k}X_{k,j}$$\n",
        "\n",
        "## Single vector transformation\n",
        "\n",
        "To simply things a bit, we start with the case where $x$ is a single vector.\n",
        "\n",
        "<br>$$\\mathbf{C_{(m*1)}} = A_{(m*p)}x_{(p*1)}$$ <br>\n",
        "\n",
        "We can think of a matrix $\\mathbf{A}$ as a **transformation** that acts on a $p$-dimentional vector $x$ by multiplication to produce a $m$-dimentional new vector $\\mathbf{A}x$. We will use python to visualize these transformations.\n",
        "\n",
        "**Helper function to visualize single vector transformation**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "MXcX3VnIuXwm",
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      "source": [
        "def viz_single_trans(orignal_data, scaled_data, title=''):\n",
        "  origin = [0], [0]   # origin point\n",
        "  f, ax = plt.subplots(figsize=(6, 5))  \n",
        "  x, y = orignal_data[0], orignal_data[1] \n",
        "  t_x, t_y = scaled_data[0], scaled_data[1] \n",
        "  ax.quiver(*origin, x, y, \n",
        "            color='b', scale_units='xy', \n",
        "            width=0.01, angles='xy', scale=1)\n",
        "  ax.quiver(*origin, t_x, t_y, \n",
        "            color='r', scale_units='xy', \n",
        "            width=0.01, angles='xy', scale=1)\n",
        "  ax.text(x + 0.1, y + 0.1, \"$\\mathbf{x}$\", color='b', fontsize=14)\n",
        "  ax.text(t_x + 0.1, t_y + 0.1, \"$\\mathbf{t}$\", color='r', fontsize=14)\n",
        "  ax.set_xlim([-2, 2])\n",
        "  ax.set_ylim([-2, 2])\n",
        "  ax.set_aspect('equal')\n",
        "  ax.axhline(y=0, color='k')\n",
        "  ax.axvline(x=0, color='k')\n",
        "  ax.set_title(title)\n",
        "  plt.show()"
      ],
      "execution_count": 1,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "FbNbSohTuZ6H",
        "colab_type": "text"
      },
      "source": [
        "**Case 1: Rotation Matrix** \n",
        "\n",
        "$$A = \\begin{bmatrix}\n",
        "cos(\\theta) & -sin(\\theta) \\\\\n",
        "sin(\\theta) & cos(\\theta)\n",
        " \\end{bmatrix}$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "JkMk0E9eRsHO",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 331
        },
        "outputId": "f44caf8e-3c17-4040-b5c6-2c0b7f81cc7f"
      },
      "source": [
        "# orignal vector\n",
        "x = np.array([1, 0])  \n",
        "# 45 degrees in radian\n",
        "theta = (45*np.pi)/180   \n",
        "# Rotation matrix for 45 degrees\n",
        "A = np.array([[np.cos(theta), -np.sin(theta)],\n",
        "              [np.sin(theta), np.cos(theta)]]) \n",
        "# the rotated vector\n",
        "Ax = np.matmul(A, x)\n",
        "viz_single_trans(x, Ax, title='Rotation Transformation')"
      ],
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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\n",
            "text/plain": [
              "<Figure size 432x360 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ocYk_dg9UTCI",
        "colab_type": "text"
      },
      "source": [
        "In this example, the multiplication with matrix A rotates the original vector $(1,0)$ by 45 degree (counterclockwisely) and its magnitude remain unchanged.\n",
        "\n",
        "**Case 2: Streching Matrix in x-direction**\n",
        "\n",
        "$$B = \\begin{bmatrix}\n",
        "m & 0 \\\\\n",
        "0 & 1\n",
        "\\end{bmatrix}$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "g20rPKgNSDFM",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 331
        },
        "outputId": "d29e1cb0-f3b2-4e03-90f7-02f60a071af2"
      },
      "source": [
        "x = np.array([1, 0])  \n",
        "B = np.array([[2, 0], \n",
        "              [0, 1]]) \n",
        "# the stretched vector\n",
        "Bx = np.matmul(B, x)  \n",
        "viz_single_trans(x, Bx, title='Stretching in x-diretion')"
      ],
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x360 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "JVY9YhGFgc6a",
        "colab_type": "text"
      },
      "source": [
        "In this example, the multiplication with matrix $\\mathbf{B}$ does not change the direction of vector $(1, 0)$ but only stretches it. This is the same with multiplying the vector with a scaler value. In this example, we can simply represent the transformation as:\n",
        "\n",
        "<br> $$Bx = 2x$$ \n",
        "\n",
        "**Case 3: Streching Matrix in y-direction**\n",
        "\n",
        "\n",
        "$$C = \\begin{bmatrix}\n",
        "1 & 0 \\\\\n",
        "0 & n\n",
        "\\end{bmatrix}$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "0x2R_xTnUp8u",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 331
        },
        "outputId": "5fcafc5b-99f7-4ef1-bf80-2463e81d4c0e"
      },
      "source": [
        "x = np.array([0, 0.5])  \n",
        "C = np.array([[1, 0], \n",
        "              [0, 3]]) \n",
        "# the stretched vector\n",
        "Cx = np.matmul(C, x)  \n",
        "viz_single_trans(x, Cx, title='Stretching in y-diretion')"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x360 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "IN8X2J25hviZ",
        "colab_type": "text"
      },
      "source": [
        "Similarly, in the case above the matrix $\\mathbf{C}$ does not change the direction of vector $(0, 0.5)$ but only stretches it. Again, it can be represented as:\n",
        "\n",
        "$$Cx = 3x$$ <br>\n",
        "\n",
        "However, this is not always the case. Check the following examples:\n",
        "\n",
        "**Case 4: Both Streching and Rotation**\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "opDVr9J8gluJ",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 331
        },
        "outputId": "27ac5e91-fa02-4b8b-83eb-4b0319a8f358"
      },
      "source": [
        "x = np.array([1, 0.5])  \n",
        "Bx = np.matmul(B, x)  \n",
        "viz_single_trans(x, Bx, title='Both roation and stretching')"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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kF6KTkxO2b98OLy+vZvPKysrg4eGBfv36QalUIiwsDLm5uR0KSkRkb2rZK6rVUKutr15VVQWNRmOe1mg0KCsrs7qsTqeTG8Fmrly5whxdKIfBYIDRaORYMIfDyS5EW/Hz8+vsCNDpdMzRhXI4OzvDYDBwLJijRQUFBXbZrl3eZfby8oJerzdPV1ZWWj21JiLqSuxSiL6+vqitrcWZM2fQ2NiI7OxshISE2GNXREQ2I/uU+dtvv8WaNWtQXl4OtVqNrKwsREREwNfXF1FRUVi2bBlSUlIAABMnTsSAAQNsFpqIyB5kF+LQoUORnp7e4vzg4GDs3btX7uaJiByOn1QhIpKwEImIJCxEIiIJC5GISMJCJCKSsBCJiCQsRCIiCQuRiEjCQiQikrAQiYgkLEQiIgkLkYhIwkIkou5pxgxAoQDuvvumV2EhEhFJWIhE3cCHH5oOhhQK4LPPTM99/jmgVJqey8jo3HwOd/fdwK5dpsenTv06ODk5ra7GQiTqBiZNAubONT3+4x+Bs2eBmTMBIUxnjvHxnRrP8UaMAG6/3fTYyQkYNcr04+7e6mosRKJuYsMGYPBg4ORJ4IEHgJ9+AgYOBDZt6uxkneAf/wAeecT0uF8/4OhR009gYKursRCJuglnZ+C99wCVCqisNJ0uv/ce4Ora2cl+O1iIRN3ImTNAU5PpsdFoOkqkm8dCJOomzp4FZs82PR4xwvTnU08BZWWdl6lTOTub/jQYTBdTbwILkagbuPbmiV4PBAcDubmm9xAuXgSmTzcdLd5y7r/f9GdVlenx6NFAXV2rq7AQibqBTZuArCygZ0/T3SY9ewJ//avpICk7G1i/vrMTdoL/+R9gyhTAwwMoKQHy8n69ntAC2b+GlIi6jgULTD/Xu+8+4PLlzsnTJbi6ApmZ7VqFR4hERBIWIhGRRPYp88qVK1FUVASFQoElS5YgICDAPC8iIgI+Pj5QqVQAgHXr1sHb27vjaYmI7EhWIR47dgynTp3C3r17ceLECSxZsgR79+61WGb79u1wcXGxSUgiIkeQVYi5ubmIjIwEAAwcOBDV1dWora2FK2+JJ6L2+L//A+rrTW+AXPvp1cv0RQydQFYh6vV6+Pv7m6c1Gg2qqqosCjE1NRXl5eUICgpCSkoKFC28QJ1OJyeCTV25coU5ulAOg8EAo9F4y4/FxYtK/PhjT5w40RPh4Q0Aut949PrmG9w1YwaUV66YnxNKJYy9e8Po7AyjszOE9Oe1n/q77zZ9g4Ud2OS2G3HDXeALFizA2LFj4eHhgeTkZGRlZUGr1Vpd18/PzxYROkSn0zFHF8rh7OwMg8Fwy4yFwQDodMDx48C33/76Z0WF6Yta9u4FvL0vds/x8PMD1GogLs78lMJohOryZahuvGdIrQb+9Cfgz3/GqZIS22W4fhdyVvLy8oJerzdPnzt3Dp6enubp2NhY8+PQ0FCUlJS0WIhEt5KzZ4HDhy2L78cfrX+yzNkZOHAAiIw0FWa3Uldn+jhNTo7pR6ls/eM0kZGmu8+vffrETmQVYkhICDZt2oSEhAQUFxfDy8vLfLpcU1ODZ599Fm+99RacnJyQn5+P6Ohom4Ym+q1ycTF9M9X777e+XJ8+wMcfA2PGOCaX3d1YgHl5pmuHbbnzTuD114HHH3fIdUVZhRgYGAh/f38kJCRAoVAgNTUV+/btg5ubG6KiohAaGoqpU6eiZ8+eGDJkCI8OiSRubqav5OrfH1i71voyXl7Ap58Cw4c7NptNtacAlUrgnntMh8rXODkBCxcCixf/+iUNDiD7GuLzzz9vMX3/dYey06dPx/Tp0+WnIuqmSkqAFStMpWhN//6mXwFw332OzdVh7S3AwEAgPNz08/DDwMGDQEKCaf6jj5qOCu+91zHZr8PPMhM5wPVF2NKlskGDTGV4552OzSZLRwvQw8Nymdxc09d7b9xoKsROwkIksqOWirB/f2DJEmDpUuCXX4CAANNpcpf9QJdUgLdnZpreCepoAd4oIgJYvdp0D2InYiES2UFbRThzpukrupYtM31N38cfA7fd1mlxm2vhCNDT2rJyCvBGkyZ1NLFNsBCJbOhmi/Ca8eOBrVu7wO89sfUp8G8UC5HIBtpbhNf89a+mXwrlcDILsGzgQPSfNq3bFOCNWIhEHSC3CK9xWBna6AiwVqfrtmUIsBCJZOloEdodT4FlYSEStUOXLUIWoE2wEIluQpcrQhagXbAQiVpx8qQTVq3q/CJUXLkCHDrEArQzFiKRFb8eEd7TOUV4wxHgfUePAg0N1pdlAdoMC5HoOs1PjU3fsGL3ImzjFNjit8GxAO2GhUiElq8R+vg0IDW1h+2LsJ3XAOuGDEFvrZYFaGcsRLqltfVmyZgxJzB8uA2+lLSDb4KcrKjoEt+Y3d2xEOmWdLPvGut0Vr7K+mbY+l3gigp5OahdWIh0S7Hb7TO8DaZbYCHSLcHmRcgC7JZYiNSt2awIWYC3BBYidUsdLkJ7fyEqdUksROpWZBeho78QlbokFiJ1C+0uQp4CkxUsRPpNu+kirKsDDvELUal1LET6TWqzCBPq0LMwF3g1h1+ISjeNhUi/KS0V4b3/rw7r43Ix0TkH6r/lAM/wFJjaj4VIvwk3FmEv1GEMcjHJPQdPeOWg3+k8KN5gAVLHyC7ElStXoqioCAqFAkuWLEFAQIB53pEjR7BhwwaoVCqEhoYiOTnZJmHp1nOtCD/YXYdRIhepyEE4cjAKeeiJeuASTD/XYwGSTLIK8dixYzh16hT27t2LEydOYMmSJdi7d695/ooVK5CWlgZvb28kJiYiOjoa9957r81CU/d3xSCwKuoQGj/LwWzkYPu1ArSGBUg2IqsQc3NzERkZCQAYOHAgqqurUVtbC1dXV5SVlcHDwwP9+vUDAISFhSE3N7fFQgwPD5eX3IYMBgOcnZ07OwZzSP773/+isboWWcXjAQD/sbaQqyvQp4/px8MDUKuB/HzTz2uv2SxLZ48Fc1i3fv16u2xXViHq9Xr4+/ubpzUaDaqqquDq6oqqqipoNBqLeWVlZS1uy2AwyIlgU0ajkTm6UA6j0QgoVYCx6dfnnJ3R5O4Oo5sbmtzcLH9/Z319y2+g2CAL/066Xg57scmbKkLI/IokmE6/O5tOp+sS3zXHHCbh4eG4fLYKb9w5EUOeDkePcZ13CtzZY8Ec1hUUFNhlu7IK0cvLC3q93jx97tw5eHp6Wp1XWVkJLy+vDsakW43C3QXDP7XdqS/RzVC2vUhzISEhyMrKAgAUFxfDy8sLrq6uAABfX1/U1tbizJkzaGxsRHZ2NkJCQmyXmIjITmQdIQYGBsLf3x8JCQlQKBRITU3Fvn374ObmhqioKCxbtgwpKSkAgIkTJ2LAgAE2DU1EZA+yryE+//zzFtP33//r750IDg62uA2HiOi3QNYpMxFRd8RCJCKSsBCJiCQsRCIiCQuRiEjCQiQikrAQiYgkLEQiIgkLkYhIwkIkIpKwEImIJCxEIiIJC5GISMJCJCKSsBCJiCQsRCIiCQuRiEjCQiQikrAQiYgkLEQiIgkLkYhIwkIkIpKwEImIJCxEIiIJC5GISKKWs1JDQwMWLVqEiooKqFQqrFq1Cv3797dYxt/fH4GBgebpnTt3QqVSdSwtEZEdySrEjz76CO7u7li/fj2++OILrF+/Hhs3brRYxtXVFenp6TYJSUTkCLJOmXNzcxEVFQUAeOihh1BYWGjTUEREnUHWEaJer4dGowEAKJVKKBQK1NfXw8nJybxMfX09UlJSUF5ejujoaMycOdPqtnQ6nZwINnXlyhXm6EI5DAYDjEYjx4I5HK7NQszIyEBGRobFc0VFRRbTQohm6y1cuBCTJk2CQqFAYmIiRo4ciWHDhjVbzs/Pr72ZbU6n0zFHF8rh7OwMg8HAsWCOFhUUFNhlu20WYnx8POLj4y2eW7RoEaqqqnD//fejoaEBQgiLo0MAmDZtmvnx6NGjUVJSYrUQiYi6ClnXEENCQnDw4EEAQHZ2NkaNGmUxv7S0FCkpKRBCoLGxEYWFhRg0aFDH0xIR2ZGsa4gTJ07EkSNHMG3aNDg5OWH16tUAgG3btiE4OBgjRoyAj48P4uLioFQqERERgYCAAJsGJyKyNVmFeO3ewxvNmTPH/PiFF16Qn4qIqBPwkypERBIWIhGRhIVIRCRhIRIRSViIREQSFiIRkYSFSEQkYSESEUlYiEREEhYiEZGEhUhEJGEhEhFJWIhERBIWIhGRhIVIRCRhIRIRSViIREQSFiIRkYSFSEQkYSESEUlYiEREEhYiEZGEhUhEJGEhEhFJWIhERBLZhXjs2DGMGTMG2dnZVud/+OGHmDJlCuLj45GRkSE7IBGRo6jlrHT69Gm8++67CAwMtDrfYDBgy5YtyMzMRI8ePRAXF4eoqCj06dOnQ2GJiOxJ1hGip6cnNm/eDDc3N6vzi4qKMGzYMLi5uaFXr14IDAxEYWFhh4ISEdmbrCPE3r17tzpfr9dDo9GYpzUaDaqqqqwuq9Pp5ESwqStXrjBHF8phMBhgNBo5FszhcG0WYkZGRrNrgPPnz8fYsWNveidCiBbn+fn53fR27EWn0zFHF8rh7OwMg8HAsWCOFhUUFNhlu20WYnx8POLj49u1US8vL+j1evP0uXPn8MADD7Q/HRGRA9nltpvhw4fj+PHjuHTpEi5fvozCwkKMHDnSHrsiIrIZWdcQc3JykJaWhtLSUhQXFyM9PR07duzAtm3bEBwcjBEjRiAlJQWzZs2CQqFAcnJyi2/AEBF1FbIKMTw8HOHh4c2enzNnjvmxVquFVquVHYyIyNH4SRUiIgkLkYhIwkIkIpKwEImIJCxEIiIJC5GISMJCJCKSsBCJiCQsRCIiCQuRiEjCQiQikrAQiYgkLEQiIgkLkYhIwkIkIpKwEImIJCxEIiIJC5GISMJCJCKSsBCJiCQsRCIiCQuRiEjCQiQikrAQiYgkLEQiIonsQjx27BjGjBmD7Oxsq/P9/f2RlJRk/mlqapIdkojIEdRyVjp9+jTeffddBAYGtriMq6sr0tPTZQcjInI0WUeInp6e2Lx5M9zc3Gydh4io08g6Quzdu3eby9TX1yMlJQXl5eWIjo7GzJkzrS6n0+nkRLCpK1euMEcXymEwGGA0GjkWzOFwbRZiRkYGMjIyLJ6bP38+xo4d2+p6CxcuxKRJk6BQKJCYmIiRI0di2LBhzZbz8/NrZ2Tb0+l0zNGFcjg7O8NgMHAsmKNFBQUFdtlum4UYHx+P+Pj4dm942rRp5sejR49GSUmJ1UIkIuoq7HLbTWlpKVJSUiCEQGNjIwoLCzFo0CB77IqIyGZkXUPMyclBWloaSktLUVxcjPT0dOzYsQPbtm1DcHAwRowYAR8fH8TFxUGpVCIiIgIBAQG2zk5EZFOyCjE8PBzh4eHNnp8zZ4758QsvvCA7FBFRZ+AnVYiIJCxEIiIJC5GISMJCJCKSsBCJiCQsRCIiCQuRiEjCQiQikrAQiYgkLEQiIgkLkYhIwkIkIpKwEImIJCxEIiIJC5GISMJCJCKSsBCJiCQsRCIiCQuRiEjCQiQikrAQiYgkLEQiIgkLkYhIwkIkIpKwEImIJGo5KzU2NuLFF1/E6dOn0dTUhIULF2LkyJEWy3z44YfYtWsXlEolnnjiCcTHx9skMBGRvcgqxAMHDqB37954//338cMPP2Dx4sXIzMw0zzcYDNiyZQsyMzPRo0cPxMXFISoqCn369LFZcCIiW5N1yjxp0iQsXrwYAKDRaHDx4kWL+UVFRRg2bBjc3NzQq1cvBAYGorCwsONpiYjsSNYRYo8ePcyPd+3ahUcffdRivl6vh0ajMU9rNBpUVVVZ3VZBQYGcCDbHHJY6M8f69es7PcP1mMNSV8lhD20WYkZGBjIyMiyemz9/PsaOHYv33nsPxcXFePvtt1vdhhDC6vNBQUHtiEpEZF9tFmJ8fLzVN0QyMjJw6NAhvPnmmxZHjADg5eUFvV5vnj537hweeOABG8QlIrIfWdcQy8rKsGfPHmzevBk9e/ZsNn/48OE4fvw4Ll26hMuXL6OwsLDZu9BERF2NrGuIGRkZuHjxIubMmWN+Li0tDTt37kRwcDBGjBiBlJQUzJo1C93GjkgAAAUJSURBVAqFAsnJyejduzf+93//t0vcqnPs2DE888wzWLlyJcaNG9dsvr+/PwIDA83TO3fuhEqlcngOR4xHQ0MDFi1ahIqKCqhUKqxatQr9+/e3WMbe47Fy5UoUFRVBoVBgyZIlCAgIMM87cuQINmzYAJVKhdDQUCQnJ9tsv+3JERERAR8fH/PrXrduHby9ve2So6SkBE8//TRmzJiBxMREi3mOHI/WcjhyPNauXYuCggI0NjZi7ty5mDBhgnmezcdDOEhmZqZITU0VQghRUlIipkyZYjH/8uXLYsKECeLSpUuirq5OPPLII+LChQs2z3Hq1Ckxb9488fTTT4tDhw5ZXebBBx+0+X7bm8NR47Fv3z6xbNkyIYQQhw8fFs8880yzZew5Hnl5eWLOnDlCCCF+/PFH8cQTT1jMj4mJERUVFaKpqUlMmzZN/PDDD52SY9y4caK2ttYu+77e5cuXRWJioli6dKlIT09vNt9R49FWDkeNR25urpg9e7YQQojz58+LsLAwi/m2Hg+HfVKlq9yq4+npic2bN8PNzc3m27ZlDkeNR25uLqKiogAADz30kMNvj8rNzUVkZCQAYODAgaiurkZtbS0A06UZDw8P9OvXD0qlEmFhYcjNzXV4DkdycnLC9u3b4eXl1WyeI8ejtRyOFBwcjDfeeAMA4O7ujrq6OjQ1NQGwz3g4rBB79Ohhvt7Y0Vt1OqJ3795tnu7V19cjJSUFCQkJePfdd22e4WZyOGo8rt+PUqmEQqFAfX29xTL2HA+9Xo/bbrvNPH3966yqqnLIGLSV45rU1FRMmzYN69ata/HOiY5Sq9Xo1auX1XmOHI/WclzjiPFQqVRwdnYGAGRmZiI0NNT8/8Ye4yHrGmJb7Hmrjq1ytGbhwoWYNGkSFAoFEhMTMXLkSAwbNszhOa5nr/EoKipqcz+2Ho/W2Os/VnvdmGPBggUYO3YsPDw8kJycjKysLGi12k5K1/kcPR6fffYZMjMzsWPHDrvtA7BTIXaVW3VaytGWadOmmR+PHj0aJSUlHSoAOTkcNR6LFi1CVVUV7r//fjQ0NEAIAScnJ4tlbD0e17P2Oj09Pa3Oq6ystNspXGs5ACA2Ntb8ODQ0FCUlJQ4vREeOR1scOR6HDx/G22+/jXfeecfiEpM9xsNhp8y/lVt1SktLkZKSAiEEGhsbUVhYiEGDBjk8h6PGIyQkBAcPHgQAZGdnY9SoURbz7T0eISEhyMrKAgAUFxfDy8sLrq6uAABfX1/U1tbizJkzaGxsRHZ2NkJCQmy275vNUVNTg1mzZpkvJeTn53fKvwlHjkdrHDkeNTU1WLt2LbZu3drsuxDsMR4K4aBzlA0bNuBf//oX7rjjDvNzN96qc/DgQaSlpZlPzSZNmmTzHDk5OUhLS0NpaSk0Gg08PT2xY8cObNu2zZzjtddew9GjR6FUKhEREYGnnnqqU3I4YjyampqwdOlSnDx5Ek5OTli9ejX69evn0PFYt24dvvrqKygUCqSmpuK7776Dm5sboqKikJ+fj3Xr1gEAJkyYgFmzZtl03zebY9euXdi/fz969uyJIUOG4M9//jMUCoXNM3z77bdYs2YNysvLoVar4e3tjYiICPj6+jp0PNrK4ajx2Lt3LzZt2oQBAwaYnxs1ahQGDx5sl/FwWCESEXV1/IJYIiIJC5GISMJCJCKSsBCJiCQsRCIiCQuRiEjCQiQikrAQiYgk/x8XgcT41SbCggAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x360 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ErdyDXFDiUAx",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 331
        },
        "outputId": "2832f54d-b70d-41eb-826e-c149a82db108"
      },
      "source": [
        "x = np.array([1, 0.5])  \n",
        "Cx = np.matmul(C, x)  \n",
        "viz_single_trans(x, Cx, title='Both roation and stretching')"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x360 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "shRv7Q9Lgtd3",
        "colab_type": "text"
      },
      "source": [
        "In these two examples, we apply the same matrixes $\\mathbf{B}$ and $\\mathbf{C}$ to a different vector $(1, 0.5)$, it turns out change both the direction and the magnitude of this vector. This is because the general effect of matrix on the vectors is a **combination of rotation and stretching**. However, for each matrix $\\mathbf{M}$, there exists some vectors who have this property that multipling with $\\mathbf{M}$ have the same effect with multipling an scaler $\\lambda$: \n",
        "\n",
        "$$Mx = \\lambda x$$ <br>\n",
        "\n",
        "\n",
        "These vectors $x$ are called the **eigenvectors** of $\\mathbf{M}$, and their corresponding scalar quantity $\\lambda$ is called an **eigenvalue** of $\\mathbf{M}$ for that eigenvector. This idea can be better understood with groups of vectors transformation below.\n",
        "\n",
        "## Transformation of group of vectors\n",
        "\n",
        "Now, instead of applying transformation on a single vector, we can apply on a group of vectors. \n",
        "\n",
        "Here I will be using the unit circle defined by $(cos(\\theta), sin(\\theta))$ where $\\theta \\in [0, 2\\pi]$ for illustration.\n",
        "\n",
        "**Helper function to visualize group transformation**"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Ix_YNSrkDZFX",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "def viz_group_trans(orignal_data, scaled_data):\n",
        "  origin = [0], [0]   # origin point\n",
        "  fig = plt.figure(figsize=(10, 15)) \n",
        "  def _plot(ax, x1, y1, x2, y2, isorigin):\n",
        "    label = 'x' if isorigin else 't'\n",
        "    color = 'b' if isorigin else 'r'\n",
        "    ax.quiver(*origin, x1, y1, \n",
        "              color=color, scale_units='xy', \n",
        "              width=0.01, angles='xy', scale=1)\n",
        "    ax.quiver(*origin, x2, y2, \n",
        "              color=color, scale_units='xy', \n",
        "              width=0.01, angles='xy', scale=1)\n",
        "    if isorigin:\n",
        "      ax.plot(orignal_data[:,0], orignal_data[:,1])\n",
        "    else:\n",
        "      ax.plot(scaled_data[:,0], scaled_data[:,1])\n",
        "    ax.set_title(\"After transformation\")\n",
        "    ax.text(x1 + 0.2, y1 + 0.1, \"$\\mathbf{\"+label+\"_1}$\", color=color, fontsize=14)\n",
        "    ax.text(x2 + 0.2, y2 + 0.1, \"$\\mathbf{\"+label+\"_2}$\", color=color, fontsize=14)\n",
        "    ax.set_aspect('equal')\n",
        "    ax.axhline(y=0, color='k')\n",
        "    ax.axvline(x=0, color='k')\n",
        "    ax.set_xlim([-4, 4])\n",
        "    ax.set_ylim([-4, 4])\n",
        "\n",
        "  ax1 = fig.add_subplot(1, 2, 1)  \n",
        "  x1, y1 = orignal_data[0, 0], orignal_data[0, 1]  # random example 1\n",
        "  x2, y2 = orignal_data[90, 0], orignal_data[90, 1]  # random example 2\n",
        "  _plot(ax1, x1, y1, x2, y2, isorigin=True)\n",
        "  ax1.set_title(\"Before transformation\")\n",
        "\n",
        "  ax2 = fig.add_subplot(1, 2, 2)  \n",
        "  x1, y1 = scaled_data[0, 0], scaled_data[0, 1]  # transformed example 1\n",
        "  x2, y2 = scaled_data[90, 0], scaled_data[90, 1]  # transformed example 2\n",
        "  _plot(ax2, x1, y1, x2, y2, isorigin=False)\n",
        "  ax2.set_title(\"After transformation\")\n",
        "  \n",
        "  plt.show()"
      ],
      "execution_count": 5,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "lCK7hxauZ_rg",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "# Generate data on a unit circle\n",
        "x = np.cos(np.linspace(0, 2*np.pi, num=360))\n",
        "y = np.sin(np.linspace(0, 2*np.pi, num=360)) \n",
        "orignal_data = np.column_stack([x, y])"
      ],
      "execution_count": 6,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "zj5kovCDDHj2",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 313
        },
        "outputId": "d5e82cfd-71e4-40ac-9671-edb1b02f35c8"
      },
      "source": [
        "A = np.array([[3, 2],\n",
        "              [0, 2]]) \n",
        "scaled_data = np.matmul(A, orignal_data.T).T\n",
        "viz_group_trans(orignal_data, scaled_data)"
      ],
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 720x1080 with 2 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "m_DJpbcln8ug",
        "colab_type": "text"
      },
      "source": [
        "The sample vectors $x_1$ and $x_2$ on the unit circle are transformed into $t_1$ and $t_2$ respectively, namely:\n",
        "\n",
        "<br>$$t_1 = \\mathbf{A}x_1 \\qquad  t_2 = \\mathbf{A}x_2$$<br>\n",
        "\n",
        "From this example you can see that the multiplication with matrix $\\mathbf{A}$ only changes the magnitude of vector $x_1$, but changes both of the direction and magnitude of vector $x_2$. This means $x_1=(1,0)$ is an eigenvector of matrix $\\mathbf{A}$ with eigenvalue 3, but $x_2=(0,1)$ is not."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "OvmVO1NdNjzj",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 313
        },
        "outputId": "61a417cc-55eb-4ba2-a8a3-9b6c55732695"
      },
      "source": [
        "A = np.array([[1, 0],\n",
        "              [0, 3]]) \n",
        "scaled_data = np.matmul(A, orignal_data.T).T\n",
        "viz_group_trans(orignal_data, scaled_data)"
      ],
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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UlITk5GR4eHjctGaeRCLBokWL8P3332PUqFF444038MEHH9Q5BHenxo4di9LSUowcORJPP/00nnjiCeTn52PBggV1avKNhxqTk5MxZMgQTJo0CWPHjkVkZCTuv/9+i3OgOycRbjXEQEREREQ34YgTERERkZka1ThpNBokJiZi9erV1sqHiMhuWMOI6E41qnH69NNP7+iUUSIiR8IaRkR3yuLG6fz58zh37lyD6+MQETki1jAisoTFjVPt2UpERM6INYyILGHRAphr165F9+7dERMTc9ttMjIyLE6KiJxX7eVzHBlrGBHdijn1y6LGafv27cjJycH27duRn58PT09PREZG1lll1dwEbCUzM/Omaw25S3wxYyckJEClUmH//v2ixAfc97V3hPjO0mw4eg0T+3101/juXr/Eji/2cze3flnUOL3//vum24sXL0azZs1uKjhERI6KNYyILMV1nIiIiIjM1OiL/P71avdERM6ENYyI7gRHnIiIiIjMxMaJiIiIyExsnIiIiIjMxMaJiIiIyExsnIiIiIjMxMaJiIiIyExsnIiIiIjMxMaJiIiIyExsnIiIiIjMxMaJiIiIyExsnIiIiIjMxMaJiIiIyExsnIiIiIjMxMaJiIjoTjzwACCRAC1aiJ0JiYCNExEREZGZ2DgRERGZq0ULYOlS4+1Ll4wjTxIJsH27mFmRHcnFToCIiMhpxMcDVVVAURHg6Wn8GgACAsTNi+yGI05ERETmWrMGGDPGeDsqCkhLM/7z9weGDQNCQ423x4wB8vLEzZVsgo0TERFRY+XlAYIAvPYaMGMGsHEj8PjjYmdFNsBDdURERI01YEDdeU7LlgEnToiWDtkOR5yIiIjuhEJh/F+lMo4yAcb5TrX27DHOgxo0yP65kc2xcSIiIroTHToY/y8sNN7u1w9Qq433ZWYCU6YY71+4ULwcyWZ4qI6IiOhOzJ4N7NwJbN0KnDljvE+vNx6aGz7cODl8yxYgJETcPMkmLGqc1Go15s+fj+LiYlRXV+PRRx/FsGHDrJ0bEZFNsIZRo/j5AatW1b0vJ8d4Vp1SCTzzDLB7N+DrC4wbJ06OZDMWNU7btm1D586dMWfOHOTl5WH27NksOkTkNFjDyOrOnzceugOA554z/t+8ORsnF2RR4zR69GjT7StXrqBJkyZWS4iIyNZYw+hGGq0eZwsqUVxVg2qNHptO5KN1uB9ahvlCJpWYt5OEhOsTxcmlNWqOU0pKCvLz87FkyZJbfj8zM7Mxu28UjUbjtvHFjK1SqWAwGPjai0Ts+M7GUWuY2O+jO8Q3CALS89T49Uw5Dl1Wo1ovIL+gAgDwyLIMAECAlxQDYn0xpn0A2oR62TSfWu7w2jti7DvRqMbpxx9/RGZmJp599lmsX78eEkndzjwuLq5RyTVGZmam28YXM7ZCoYBKpeJrLxKx42dkZIgW2xKOWsPEfh9dPf6Bi0r8c+1xnMqvQJMAL6T0iUW/VqH411YFDLk5+PqxgThTUIldZwux9WQBfjtbgcS4JvjXuI6ICVHYLC/A9V97R40NmF+/LGqcjh8/jtDQUERFRSEuLg56vR5KpRKhoaGW7I6IyK5Yw9xTjc6A//yaiW/2XESzIB8suqcbxnVrCg+ZcWWehRWlEJRF6NosEF2jgzClZzTKNVos3XMRS3acR9L7O/HK2I5I6RMr8jMhMVm0jlN6ejq+/vprAEBRURFUKhWCg4OtmhgRka2whrkfZVUN7vsyDd/suYgHBrTAlqeGYFKPaFPTBAAoLoZEqwWOHDHdFeDtgXkj2mLTk0MQHxuE+auP4Z9rj0OnN4jwLMgRWNQ4paSkQKlU4t5778XcuXPxyiuvQCrlWppE5BxYw9zL1XINpn22F0dzy/Dh9Hi8Or4TFJ5/OeCiVgMlJcbbP/980z6igxX4bnZfzB3SCsvSLuHxHw9Dy+bJLVl0qM7b2xvvvvuutXMhIrIL1jD3UaqqwfQv0nClTINvHuyNAa3Dbr3hH38AhmuN0M8/A//8502byKQSvDg6DhH+XnjjF+Mk5sXT4yE198w7cglcOZyIiFySRqvHnO/SkaNU47uH+qBfq3rmsG3YcP32gQPAlStAVNQtN314cCsYBAFvbTyFZsE+eHG0eBOayf44Nk1ERC7p1fUncOBiCd69p1v9TZMg1G2cgJu//os5g1vh/v7N8fnOC1iVkWuFbMlZsHEiIiKXs/pgLn48kIP/G9Ya47o1rX/jI0eA3L80P7eY53QjiUSCV8Z2RL9WIXhl3XFcKKxsZMbkLNg4ERGRS8ktUeHltcfRp2UInkxs1/ADbjW6tHWrccJ4PeQyKd6b1h2ecike//EwanScLO4O2DgREZHLEAQBL605DgBYdE83yGVm/Jq71eiSWg38/nuDD40K9MHCyV1xLK8MH207d6fpkhNi40RERC5j7eE87DhTiOeS2iM62IxVvgsKgP37AcUN2/r6Gv9v4HBdraROkRjfrSmW7DiPi0VVFmRNzoSNExERuYRyjRavb8hEj9ggzOzfwrwH7dwJPP88cOHC9fseewzYssW4rpPBvMNvL4+Jg6dMin+tPwGBF/t1aWyciIjIJXy24zyUVTX49/jOkJm7ttKUKcCCBUB4+PX7JBIgMRFYscJ42wwRAd54cmQ77DhTiC0nCyzInpwFGyciInJ6V8rU+HJXFiZ0b4ou0YHmP/BWjVHtiJFEYnbjBACz+jdHqzBfvLP5NPQGjjq5KjZORETk9N7bcgaCADxzV3vLdnAHDdLtyGVSPDGyHc4UVGLD0cuN3h85JjZORETk1HJLVPjpYB7u7RuLmBAzJoTb0NguUWjfxB/vbz3LCwG7KDZORETk1L7clQWpBJg7pJV1dtiIyd1SqQRP3dUOWUVV+OXYFevkQw6FjRMRETmt4spq/HggGxO7N0PTIB/Ld2SFQ3W1RsY1QatwX3y1O4tn2LkgNk5EROS0lu69hGqdAY8MtdJoE9CoESfAOOr00KCWOJpbhgMXS6yUFDkKNk5EROSUtHoD/rcvG8PaR6BNhL/Y6dQxKT4aQQoPfLnrQsMbk1Nh40RERE5py8kCFFVWY0a/WLFTuYmPpwz39Y3FlswC5JXWf807ci5snIiIyCkt33cJzYJ8MLRdhHV3bKV5SSm9jQ3dqvRcq+yPHAMbJyIicjpZRVXYc64Y0/vEmL9KuJ3FhCgwsHUYUjNyYOCCmC6DjRMRETmd1QdzIZUAU3vFiJ1Kvab2ikZuiRp7LxSLnQpZCRsnIiJyKoIgYN3hyxjQOgxNArxtEcBqu0rqFIlAHw+sTM+x2j5JXGyciIjIqRzOKUW2UoUJ3ZuKnUqDvD1kGN0lCltOFkCj1YudDlkBGyciInIq6w5fhqdciqTOkbYJYOVFK8d2jYKqRo/tpwutul8SBxsnIiJyGnqDgA1Hr2BEhwgEeHuInY5Z+rYMQYivJy/B4iLkjXnw22+/jYyMDOh0OjzyyCO46667rJUXEZFNsX45p8M5pSiqrMaoLlFip2I2uUyK5M6RWHsoDxqtHt4eMrFTokawuHFKS0vD2bNnsWLFCpSUlODuu+9m4SEip8D65by2nCyAXCpBQvtw2wWxwfXlxnSJwv/2ZWPHmUIkdbLRIUayC4sbp969e6Nr164AgICAAKjVauj1eshk7KSJyLGxfjmvrZkF6Ncq1GkO09Xq0zIE/l5ybD99lY2Tk7O4cZLJZFAoFACAVatWYciQITcVnczMzMZl1wgajcZt44sZW6VSwWAw8LUXidjxnYU59QsQr4aJ/T46avy8ci3OXa1EYgsvm+ZXrFTiqg323y3SC5uPX8bMDnJIJLdetNNRX3tXj30nGjXHCQC2bt2KVatW4euvv77pe3FxcY3dvcUyMzPdNr6YsRUKBVQqFV97kYgdPyMjQ7TYlqivfgHi1TCx30dHjb/n2gVzZwzriuhghc3ihwYHI9QGz398lR92rzoKBEUjrmnALbdx1Nfe1WMD5tevRp1Vt2vXLixZsgRffPEF/P0d68rURET1Yf1yPrvPFaF1uK/tmqbbjAJZS+28rG2nr9o0DtmWxY1TRUUF3n77bXz22WcICgqyZk5ERDbF+uV8anQG7LugxKA2YWKnYrEIf290bhaAHVzPyalZfKhu48aNKCkpwRNPPGG6b+HChWja1PFXciUi98b65XwOZZdArdVjoD0aJxucVVdrYOswfL0nC+oaPXw8eTKCM7K4cZo2bRqmTZtmzVyIiOyC9cv57DlXBJlUgn6tQ8VOpVH6tQ7FZzsv4GB2iX2aQLI6rhxOREQOb/e5InSNDrTPMgQ2HHHq3SIEMqkEe88X2ywG2RYbJyIicmgarR7H8srQt6VzjzYBgJ+XHF2aBSLtAhsnZ8XGiYiIHNrR3DJo9QJ6NQ8WOxWr6NcqFIdzSqGu0YudClmAjRMRETm09EtKAEBPezVONjxUBwC9mgdDZxBwLK/MpnHINtg4ERGRQ8u4WILW4b4I9vUUOxWr6B5rXALjSE6pyJmQJdg4ERGRwzIYBGRkl6BX8xCxU7GaMD8vRAf74DAbJ6fExomIiBzWJaUKpSotejS340KlNj5UBwDdYoLYODkpNk5EROSwaucBdW4WaPtgNr7kyo3iY4KQV6pGYUW13WKSdbBxIiIih3UirwyeMinaNbHj9QTtNOIEcJ6TM2LjREREDutYXhk6RPnDQ+Zav646Nw2ETCrh4Ton5FqfRCIichmCIOB4Xpl9DtPZmY+nDK3DfXEqv1zsVOgOsXEiIiKHlKNUo1yjQ+emdm6c7HCoDgDaNfHHqfwKu8Qi62HjREREDql2NCYuyo7zm+yoQ6Q/ckvUqKzWiZ0K3QE2TkRE5JDOXq0EALSJ8BM5E9toHxkAADhTwFEnZ8LGiYiIHNL5q5WICvSGv7eHfQPb6VBd+2tnCp7h4TqnwsaJ7Gr9euNSKRIJsHWr8b4dOwCp1Hhfaqq4+RGR4zh7tdJlR5sAIDrYBwpPGec5ORk2TmRX48cDjzxivD1nDpCfDzz4oPEPvAceAKZOFTU9InIQBoOAc1cr0TZChPlNdhpxkkolaNfEH6fZODkVNk5kd4sWAe3bAxcvAt27A1lZQOvWwOLFxu+fPQsMGwaEhgL+/sCYMUBenqgpE5Gd5ZWqodbq7TviZMeVw2u1jfDDucJKu8cly7FxIrtTKIDlywGZDCgoMB6mW74c8LtWH/PyjH/wvfYaMGMGsHEj8Pjj4uZMRPZV20y0beK6h+oAoEWYLworqqGq4Zl1zkIudgLknnJzAb3eeNtgMI469e1r/HrAAGD79uvbLlsGnDhh9xSJSETnr51R1zpchMbJTofqACA2RAEAyFaq0OHaWXbk2DjiRHaXnw88/LDxdny88f+//x3IyTHe9vS8vu2ePUBVFTBokH1zJCJx5ShV8PeSI1hh5zPq7KxFqC8A4GKRSuRMyFxsnMiuaieBFxUBvXsDe/caR5pKS4FZs4yjT7UyM4EpU4AOHYCFC0VLmYhEkFOiRnSIAhIR5h3ZU2xo7YhTlciZkLka1TidOXMGiYmJ+P77762VD7m4xYuBTZsALy9g6VLj/999Z5z3tG0b8O67xu1OnAASEgBfX2DLFiAkRNS0yQWxfjm2HKUK0cE+4gS346G6QB8PBCk8cLGYI07OwuI5TiqVCq+//jr69+9vzXzIxf3jH8Z/N2rXzng4rlZOjvGsOqUSeOYZYPduYwM1bpx9cyXXxfrl2ARBQG6JGoPbhoudil00D/VFNhsnp2HxiJOnpye++OILREREWDMfIpw/DxQWGiePP/ccMH06MG+e2FmRK2H9cmxlGgPUWj1iQkQacbKz5iEKXCzmoTpnYfGIk1wuh1zOk/LcUamqBoeyS3EsrwwXi6uQXaxCYWU1qqp1OJmlBAD0fH0LfDxliAzwRnSwD5qH+qJbTCC6xwQjxNez3v0nJNh1pJzcEOuXY8uv1AIAYoIV4iRg5wLULNgHG49dgcHAwucMbFo5MjMzbbn7emk0GreNb+3YgiDgnLIGey5VIS2nCpdKtabvhSlkiPL3QMsAGRQeXrgk9YBcbkDfZl5Q6wQUVanx59kKrD+iQ21NiAn0QP8YX/SLVaB9mBekVp786U5qJZAAACAASURBVEqvvbPFdzX8HIkjp0QNANCW5iMzU2m3uAIACYCS0lLk2/P5q8qgMwhIO3wcPhKd2773Yn/uzGXTxikuLs6Wu69XZmam28a3VuxyjRar0nPxfdolXCiqgkwqQZ8WIbinbxh6xAajW0wgFJ51P0KrX/SHl3cNPpk9pM79qhodjuWW4WB2KXafK8Tqk0qsPF6KFqEK3Ne3Oab0jEZwAyNR5nKF195Z42dkZIgW2xb4ORJH6vG9AIBBPTra9QK/tX/CBQcGItiOz/+SPh/YX4yAJrGQlV922/de7M+dufWLY9V0kzKVFp/tPI9v/7wIVY0e8bFBWDi5C0Z2jKz3MJteDxQXA76+spu+p/CUo2+rUPRtFYq/J7RGmUqLLZkF+HF/Nt7cmIl3t5zGrAEt8Lchra3WQBGRcypR66HwlNm1aQIgyiVXACAy0BsAkF+mQTPXXn3BJVjcOB0/fhwLFy5EXl4e5HI5Nm3ahMWLFyMoKMia+ZEd6fQGfPvnRXz4+1mUa3QY160p5g5uhS7RgWY9fv9+QKcDystlqKy8fgmVWwlUeGBKz2hM6RmNzCvl+GzHeXy+8wKWp2XjseFt8NCglvCQcZkxsg3WL8emVOkQ7u8ldhp2ExlwrXEq16CZeeWWRGRx49S5c2csW7bMmrmQiE5eLsfzPx3FsbwyDG0XjueTO6Bj0ztb/n/DBuP/gmBce+nuu817XFxUAN5Picejw9rg7d9OY8Gvp7Du8GUsnNwFXaP5i4ysj/XLsZVo9IjwF/GMOjtPDg/z84RUAhSUawA2Tg6Pf9K7OUEQ8N3ei5jw8W5cKVPjo3vj8e2Dve+4aQKAX365fnv9+jvPpV0Tf3w5qxeWzOgJZVU1Jn3yJ77anQWBp9gRuRWlSu9WI05ymRTh/l7IL9OInQqZgXOc3JhGq8cLq49hzaE8DO8QgXendrN4flFODnDkyPWvN2wwznmS3TzdqUHJnSPRv3Uonkk9gtc3nET6RSUW3dMdPp4W7IyInI5SrUOEv7fYadhVZIA38ss1ADjH09FxxMlNlWu0uP/r/Vh7OA9Pj2yHL+/v1ahJ2Rs31v26qAhIS7M8v0AfD3w+sydeGNUBv53Ix31fpqFUVWP5DonIKahr9FBpBXFHnEQY5Q7184KyijXOGbBxckPKqhpM/zwNBy+V4MOUeMwb0RZSaeNO5aid33QjSw7X3UgikeCRoa3xyb09cDyvHFOX7EVRZXXjdkpEDq2wwvgz7k6H6gAgSOGBUpW24Q1JdGyc3IyqRocHvz2As1cr8cWsXhjXrWmj96lWA7//fvP9jW2cao3qEoVvZ/dGTokKs77ejwoNiwuRqyqqMjZOYX4iHrISYcQpWOGJEo6qOwU2Tm5Epzfg/5YfxLHcUiyeHo9h7a1zna5t24zN01+dOgWcOWOVEBjQOgyf3tcTp/MrMPe7DGj1BuvsmIgcSrna+IdRoI+d13ASWbDCA6oaPWr0PBnG0bFxciOLtpzBttOFeGNiFyR1irTafjdsAPr2BR599Pp9//wn0Lkz8PPPVguDYR0i8PaUrth7oRgLfj1lvR0TkcMoc9PGKUhhHGGrqNaLnAk1hI2Tm9h26io+2X4e0/vE4N6+sVbd95w5wN69wNCh1+8bP954lt2oUVYNhUk9ovHAgBb4ancWNh67Yt2dE5HoyjU6AECAmI2TSIfqAKC8mqPpjo6NkxsoVdXgmdQj6BDpj3+N62T1/cfHG69UcOPVCgQBkEqBjh2tHg4vjo5Dt5ggvLjmGCeLE7mY2kN1Afa+3Aog2iVXAOPkcIAjTs6AjZMb+M/GUyhVa/HetO7w9rDdWkg31hyDDf9o8pRL8c6Urqiq1uH1DSdtF4iI7K5MrYWnTGLTWuWIahuncjZODo+Nk4vLuFSCFek5eHhwS8RF3flq4HfiryNOttS2iT8eTWiDdYcvY+/5YtsGIyK7KVdr4esp8q8mEQ7V1Y6wqWo4OdzRsXFyce9sOo0wPy88PqKtzWPZs3ECgL8ntEZkgDfe2Xyal2UhchFlai38xG6cRKC4dmUEjY5znByd+3063cif54uw90IxHk1oDYWn7a+uY+/GydtDhnkj2iDjUgm2ny60fUAisrkKjc4tR5xqa7RGxz8CHR0bJxf2+c4LCPf3svpZdLdj78YJAO7pFYOmgd74cvcF+wQkIpvSaPXwkok3SVss3h5SSCRmjDg98ICx2LZoYY+06BbYOLmoHKUKO84UYnrvGLtNshSjcfKQSXFfv+bYc64Y565W2icoEdmMRqeHl9z9fjVJJBL4eMg44uQE3O/T6SZWHMiBBMC0PvYZbQKMyw/UsudI9z29YuAhk2DFgewGt12//vrSCVu3Gu/bscOYu0QCpKbaOFkiqpdGa4Cn2CNOIs2ZVHjK6x9xatECWLrUePvSpevFbPt2e6RH17BxclG/Hr+C/q1D0SzIx24xxRhxAowXAx3YJgy/nchvcJL4+PHAI48Yb8+ZA+TnAw8+aMz3gQeAqVNtny8R3Z67HqoDjBPE6x1xio8HwsKMtz09jZds6NsXCLDtGdNUFxsnF5RbVoPzhVUYGdfErnHFapwAIKlTJHKUamReqWhw20WLgPbtgYsXge7dgawsoHVrYPHi69tMmQIEBRmf06uv2ixtIvoLjdYAT7kbN07aekac1qwBxowx3o6KAtLSjP969GDRsiM2Ti4oPc94xd0RbtQ4jYgzXrB459mGz65TKIDlywGZDCgoMB6mW74c8PO7vo2Xl3F0iojsq1qrd9tDdV4eMlRbepFfFi27YePkgk4XaRAV6I2YEIVd44rZOEX4e6N5qAKHskvM2j43F9BfW6DXYDCOOt1o+XLg3nutnCQRNUijE7FxEvGSKwAgl0pgsLR2smjZDRsnF3SqsBrdY4LsHlfMxgkAesQG42B2aYPb5ecDDz9svB0fb/z/738HcnJsmBwRNchgEKDVC/CSueevJplUAkNDxVNx7Q9ilUq0kTF3556fThemqtEhv1KHTk3tP1nQXtequ52OUQEorKiu9yKZtZPAi4qA3r2BvXuNcytLS4FZs8TJm4iM9NcaAanYv5lEakjkUgn0DdWgDh2M/xcWGm/36weo1TbPja4T++NJVpZXYvwBsvdhOsA2yxGcPm3+vmqfc0Gl7rbbLF4MbNpknA6wdKnx/+++M/4Rt20b8O671siaiCxRO9riMlPDs7ONI0NmkkklpubxtmbPBiZPBgIDgTNngH37rs87ILuwuHF66623MG3aNKSkpODo0aPWzIkaIbfU2DhFB9tvGYJatjhU9/nnQGys8VDaxo2ARnP7bWNCjM85v1J7223+8Q9jbhoNEBdnvK9dO6Cqynj/s88a71uxwthgAcDBg8CXXwKVXF/TpbCGOZ7auiEVea6R1QpYcTEQHg6MG2csZpcv17u5zJwRJz8/YNUq4zC5IBj/+fmxaNmRRY3T/v37cenSJaxYsQJvvvkm3nzzTWvnRRYqVdUAAEJ8vewe2xaN07PPGmvPkiXGs3BDQ4GJE401IT+/7rZBCk8AQFVN44+3Pf888P77xts//2xc86moqNG7JQfBGuaYXG7KTnw8MHIksGGDcQG5Zs2AXr2MywVkZNz0hOXmzHG6HRYtu5EIFlxW/oMPPkDTpk0x9dpqgcnJyVi1ahX8bjifOyMjA08//bT1Mr1DKpUKCoX9D1eJHf9qRTUuFFaiR2wwPO182YLSUuDIkcMAAB+f7pBb6brCVVW3n3vk729spkJDAW8fAemXlIjy90DzcHEWhHPXz12td999Fz179hQtvrkcvYaJ/T6KFd8gCNifJd7P8OGdOwFBQHe5HPCx0qh9TQ1QXX3r73l6Xi9gwcHILKhEVbUOvVqEWCe2BcT87In9uTe3fln0q62oqAidOnUyfR0SEoLCwsI6RQcwvghiMRgMbhlfU22c36PRqKGT2ne4WxAk8PQUUFMjsdtcxYoK47+LFwH/AD0QCJRq9AgX6b1318+ds3H0Gib2+yhW/Nr5PYIgiBJf8PSERKsFdDpjYbG1mhrgyhXgyhUIHh4oi2wPwH1/d4r9uTeXVcYEbjdotX//fmvs3iKZmZmIq53E4kbxl++7hJfWHMfGF0YgMtDb7vETEhJw8KAePXpst9o+Dx0Cystvvl8mAwYPNk4fGDsWaBKtRbfXNuOhniH459T+Vot/J9z1c1crIyNDtNiN4Wg1TOz3Uaz4FRotury6GXN6heClKfb/GU5ISID2/Hlsb93aeju9csU4iftW2rQxFrBx44BBg3D/skM4kavE/leSrRf/Don52RP7c29u/bKocYqIiEDRDcdOr169ivDwcEt2RVbm52V8Syurb39mma116FBttWtO5uUZL4dSKzgYGDXK2CglJxu/rpVfZjyzROHBk0Wpfqxhjql28Ucxp4Zro6Kse9HcsWOvN05SKTBokPG+ceOM1366YXKot1yKAC+Z9WKTTVj0G2bgwIHYdG32/okTJxAREXHTEDeJI8DHAwBQpr79mWXOZOFCoHlz4OmnjbXs6lXjArnTp9dtmgDgaoXxlLtgHxYeqh9rmGOSXZte4DLLqR04AOzeDUybBnz/vXHtpR07jGe9dOhw00rlBkH0xcvJDBaNOPXo0QOdOnVCSkoKJBIJ/vWvf1k7L7JQzLVlCHJLVOjZPLiBrR3fM88AH35o3raXS42NU7ivlWalk8tiDXNMHtcutaKz+LojDiY62tgseXiYtbkgCFxc0QlY/BvmmWeesWYeZCXRwQpIAGQVVYmdilXExpq/bbbS+Jyb+LFxooaxhjkej2ur6OpcZT3HqKg72lxnEMRfw4oaxObWxXh7yNDET47T+XY4I8TBHMsrR7MgH/hzjgCRU5JKJZBLJdC6yojTHarRGcS7wDGZjY2TC4qL8Eb6pZLbninkqo7llqJLs0Cx0yCiRvCQSV3nUN0dqtbpTYcryXGxcXJBHcO9UVhRjWyl46+HYS1XytS4WKxCfGyQ2KkQUSN4yCRu2zhptBxxcgZsnFxQ9yjj+k1/nLoqcib2U/tcR8RFiJwJETWGp1wKrd49G6dqnR5ecjZOjo6NkwuKDvRE+yb++PVYfsMbu4itJwsQG6JA63CeUk7kzIyH6sTOQhzVOgMP1TkBNk4ualSXSBy4pEReqZ2ufSKi/DINdp4twpiuUZDwjBQip+Ypl6JG756dUzUnhzsFNk4uakrPaEgALE+7JHYqNpeangO9QUBK7xixUyGiRvL1lEOldc9DdRqtno2TE2Dj5KKigxVIjGuCH/ZnQ6N1lUVRbqbR6rF8XzYGtglF81BfsdMhokby85ZDreWIEzkuNk4u7KFBLVGi0uJ7Fx51Wpmeg/xyDR5NaCN2KkRkBf5ecqjcsHGq1ulRozPwWptOgO+QC+vbKhSD24bho23nUK5xjWvX3UhVo8Mn286jd4tgDGgdKnY6RGQFvl7uOeJUoTFemN3Pk7+WHR3fIRf3fHIHlKq0+HDrWbFTsboPfz+H/HINnkvuwEnhRC7Cz9s95ziVX7swu4KNk8PjO+TiOjcLxPQ+sfh6TxYO55SKnY7VnCmowJe7LmBqz2j0bhEidjpEZCV+bjriVG4aceIloxwdGyc38MLoDojw98Zzq45AXeP8E8U1Wj3+8cMhBPh44IXRcWKnQ0RW5OclR7VegM7NliSoHXHy5YiTw+M75AYCvD3w9pSuOHu1Ei+tOeb017B7fcNJnMqvwLtTuyHE11PsdIjIivy85ACuz/lxF7XzUP04Odzh8R1yE0PaheOJEe2w+lAelv55Uex0LPZ92iUs35eNR4a0wrAOvLwKkaup/WNIqaoRORP7KlcbG0WOODk+vkNuZN7wNkiMa4J/bziJ9Ucui53OHdt8Ih+vrDuO4R0i8GxSe7HTISIbCPUzNk7FlW7WOGl4qM5Z8B1yI1KpBIunx6N3ixA8teIwtpwsEDsls/2eWYDHfjiELs0C8dG98ZDL+NElckWmEaeqapEzsa+Sqhp4yqXw5kV+HR5/+7gZH08ZvprVC52aBuBv32dg5YEcsVNq0Pojl/HIsgy0b+KPbx7sA4WnXOyUiMhGQn29AADFVe414lRYUY1wPy8ureIE2Di5IX9vD/xvTj8MbBOG5346inc2nYbe4HgTxg0GAYu2nME/fjiEHrHB+N+cvpwMTuTign09ALjfobrCymqE+3uJnQaZgY2Tm/L1kuOrWb0wrVcMPtp2DjO+3Ier5Rqx0zIpqqzGQ0sP4MPfz2Jyj2h891Af+Ht7iJ0WEdmYl1wGXw8plG444hTmx8bJGbBxcmMeMikWTumK/07pikM5JUh6fydS03NEXa5AEASsO5yHkYt2YM+5Yrw+oRPemdoV3h5cFI7IXQR6S1FU6V5znIo44uQ0OFmEMLVXDOJjgzD/p2N4dtVRrMrIxQuj49A9JsiueRzNLcWCX0/hz/PF6B4ThP9O6Yq2TfztmgMRiS/IW+ZWh+p0egOKq2oQ7ucJwL0W/nRGbJwIANAmwh8rH+mPlek5WPjbKUz8eA+Gd4jAowmt0bN5sM0mLAqCgIPZJfhyVxZ+PZ6PEF9P/Ht8J8zo1xwyKSdJErmjMF85chxo6oCtKVU1EARcG3FSi50ONcDiQ3X79+9H//79sW3bNmvmQyKSSiVI6ROLXc8Px7NJ7ZFxqQRTluzFqA924ds9WVadA3W1XINley9iwsd7MPnTvdh9rgj/GN4GO55NwKwBLdg0kU2xfjm2MIUcl0vVTn+VA3MVVhgPS/JQnXOwaMQpOzsb33zzDXr06GHtfMgB+HnJ8X/D2mDWgBZYf/gyftifjVd/PolXfz6JrtGBSGgXjvjYYHSNDkSomZMZS1U1OJJbhgNZSqRdKEZGdgkEAWjXxA9vTOyMu+ObwdeLA6Bke6xfji/CT45qnQHKqhqza4wzu1Jq/KM0IsAbqBI5GWqQRb+pwsPD8dFHH+Gll16ydj7kQPy85Li3byzu7RuL0/kV2JpZgC0nC7B42znU/iEY6uuJpkE+aBrkDV8vOS4UVUGv0+GpFYdRrtGhqLIaF4urUKoyroork0rQqWkAHh/RFmO6RHEOE9kd65fjC1MYfzVdLtW4ReOUW6ICAMQEK1DExsnhWdQ4+fj4mLVdZmamJbu3Co1G47bxbRV7eCQwPDIEam0QzhVX43RRNS5XaFFYVYNTeSqotQKUldUQAOw5WwCFhxQBXlIMiPZG0wB/tAr2RIdwb/h4SAHooVPmIlNp9TRd8rV3lvjOwNz6BYhXw8R+H8WOH+ShBwDsP3EW8gpfu8VVqVQwGAx2f+5HzxfDSyZBYc55VFdXu+17L/bnzlwNNk6pqalITU2tc9+8efMwePDgBnceFxdneWaNlJmZ6bbx7RH7dgc5En7/D1QqFfa9nGTT+PVx9dfekeNnZGSIFvtWGlO/APFqmNjvo9jxS9XHARRB6heKuLiWdourUCigUqns/txV6RmICdWjY8eOor/27lw/za1fDTZOU6dOxdSpUxudEBGRvbF+OadAbyk85VJcKXOPM+tyS1WIDjZ/JJTExQUwiYjIoUgkEjQL8jHN/XF1uSVqNk5OxKLGafv27Zg5cyZ27dqFRYsWYfbs2dbOi4jIJli/nEOLUAWyily/carQaFGq0iI6WCF2KmQmiyaHJyQkICEhwcqpEBHZHuuXc2gZ5oe0C0oYDAKkLryuW26JccFLjjg5Dx6qIyIih9My3BdqrR4FFa49z+lCoXH9gZZh9jt7kBqHjRMRETmcVtcaiaxC117Y6NzVSkgkQKswP7FTITOxcSIiIodTOwJzwcVXhDxXWIlmQT7w8ZSJnQqZiY0TERE5nMgAb3h7SJHl4o3T+auVaBPB0SZnwsaJiIgcjlQqQcswP1worBQ7FZsxGARcKKpEm3A2Ts6EjRMRETmkNhF+OFPguo1TXqkaGq2BI05Oho0TERE5pLgof+SVqlGm1oqdik2cu2psCluzcXIqbJyIiMghdYwKAABkXikXORPbOF1QAQBoy8bJqbBxIiIih+TqjdOJy+VoFuSDIIWn2KnQHWDjREREDinc3wthfp44edlFG6e8MnRqGiB2GnSH2DgREZFDkkgkiIsKQGa+6zVOldU6ZBVXoXOzQLFToTvExomIiBxWx6gAnMmvhFZvEDsVq8q8Ug5BAEecnBAbJyIiclgdmwagRm/AmWsTqV3FibwyAECnphxxcjZsnIiIyGHFxwQDAA5ll4qciXUdzilFuL8XmgR4iZ0K3SE2TkRE5LBiQnwQ5ueJg9klYqdiVRnZJejVPBgSiUTsVOgOsXEiIiKHJZFIEB8b7FIjTlcrNMhRqtEjNljsVMgCbJyIiMihxccGIauoCiVVNWKnYhUHLxmbwB7N2Tg5IzZORETk0GpHZlzlcN3B7BJ4yqTo3Ixn1DkjNk5EROTQukUHwVMmxb4spdipWMWBi0p0iQ6El1wmdipkATZORETk0Hw8ZYiPDcKf54vETqXRKjRaHM0tQ/9WoWKnQhZi40RERA5vQOswnLhcjjKVVuxUGmV/lhJ6g4ABrdk4OSs2TkRE5PAGtAmFIABpWcVip9Ioe84Vw0su5cRwJ2ZR46TT6fD8889j+vTpuOeee5Cenm7tvIiIbIL1yzl1iw6Cj4cMf55z7sN1f54vQq8WwfD24PwmZ2VR47Ru3Tr4+Pjghx9+wJtvvokFCxZYOy8iIptg/XJOnnIp+rQMwa6zzts4FVZU41R+BQa0DhM7FWoEuSUPGj9+PMaOHQsACAkJQWmp6yxMRkSujfXLeY2Ii8Ar607gQmElWoX7iZ3OHdt26ioAYGi7cJEzocawaMTJw8MDXl7G6+ssXbrUVISIiBwd65fzGtY+AgDwx7UGxNlszSxAVKA3OjXl+k3OrMERp9TUVKSmpta5b968eRg8eDCWL1+OEydOYMmSJbd8bGZmpnWytIBGo3Hb+GLGVqlUMBgMfO1FInZ8R9OY+gWIV8PEfh8dOX7zIA/8nJGFgWHVVo9ry/pVozdgx5mrSGztj1OnTt12O0d+7V059h0RLLRy5Uph9uzZgkajueX309PTLd21VZw8edJt44sZe+jQoULv3r1Fiy8I7vvaO0J8sX/uzdVQ/RIEcZ+L2O+jI8f/z8ZMofULvwhl6hqrx7Vl/frjVIHQ/PkNwh+nCurdzpFfe1eOLQjm/8xbdKguJycHP/74Iz766CPTkDcRkTNg/XJuIztGQGcQ8Htmgdip3JEtJwvg4yHjwpcuwKLJ4ampqSgtLcXcuXNN93311Vfw9PS0WmJERLbA+uXc4mOCERXojQ1HruDu+Gix0zGLVm/Ab8fzMSIugssQuACLGqennnoKTz31lLVzISKyOdYv5yaVSjC2axS+/fMiylRaBCo8xE6pQXvOFUFZVYNx3ZqKnQpZAVcOJyIipzK2a1No9QI2ncgXOxWz/HzkCvy95Uhoz2UIXAEbJyIicipdowMRG6LA+iOXxU6lQRqtHptP5CO5UyS85DxM5wrYOBERkVORSCS4O74Z9pwvQo5SJXY69dp8sgAV1TpM6N5M7FTIStg4ERGR07mndwwAIDU9R+RM6vfDvmxEB/tgQGueTecq2DgREZHTaRbkgyFtw7EyPRd6gyB2Ord0obASey8UY3qfWEilErHTISth40RERE4ppXcM8ss12HHGMS/BsuJADuRSCab2co5lE8g8bJyIiMgpjYhrggh/L3yz56LYqdxEXaPHyvQcJMY1QYS/t9jpkBWxcSIiIqfkKZfigYEtsOtsEU5eLhc7nTpWZeSgRKXF7EEtxU6FrIyNExEROa37+jSHwlOGL3dfEDsVE53egC92ZSE+Ngi9WwSLnQ5ZGRsnIiJyWoEKD0zrHYP1hy/jcqla7HQAAL+dyEe2UoVHhrSGRMJJ4a6GjRMRETm1hwa1hEQCLP7jnNipQG8Q8NEf59AqzBd3dWwidjpkA2yciIjIqUUHK3Bvn1isTM/BxaIqUXNZdzgPp/Ir8OTIdlyCwEWxcSIiIqf32PC28JRJsWjLGdFyqNbp8e7mM+jcLABjukSJlgfZFhsnIiJyeuH+Xpg9qAXWH7mMQ9klouSwbO8l5JWqMT85jqNNLoyNExERuYS/J7RBZIA3XlpzHDq9wa6xL5eq8d6WM0hoH45BbcPsGpvsi40TERG5BD8vOV4d3xEnr5Tj2z8v2i2uIAh4Zd0J6AUBr0/obLe4JA42TkRE5DKSOkVieIcILNpyxm4TxX89no+tmQV4MrEdYkIUdolJ4mHjRERELkMikeCNiZ3hIZNi3g+HUK3T2zReXqkaL6w+hs7NArhKuJtg40RERC6laZAP3p7SFcfyyrDw19M2i1OjM+DxHw5Bpzdg8fQe8JDxV6o74LtMREQuJ6lTJB4Y0AJf78nCqoxcq+9fEAS8vPYY0i+V4K1JXdAyzNfqMcgxsXEiIiKX9OLoOAxsE4r5Px3FzjOFVt334j/OYWV6LuYNb4MJ3ZtZdd/k2Ng4ERGRS/KUS/HpjJ5oE+GHOd+lY/vpq1bZ7+Lfz2LRljOYFN8MT41sZ5V9kvNg40RERC4rwNsD/5vTz9Q8rUzPsXhfWr0B/1x7HO9ea5r+O7UbL+LrhuSWPKi4uBjPP/88qqurodVq8cILL6Bbt27Wzo2IyOpYv9xPiK8n/vdwPzz6vww8t+ooDl4qwYtj4hDg7WH2PrKKqvDcqiM4cLEEc4e0wvPJHSDj6uBuyaIRp/Xr12PChAlYtmwZnnrqKXzwwQfWzouIyCZYv9xToMIDSx/sg0cTWmNleg4S392B5fsuQaOtf7kCZVUN3v7tFJLf34lT+RV4f1p3vDg6jk2TG7NoxOnBBx803b5y5QqaNGlitYSIiGyJ9ct9yWVSPJfcAaM6R+Gf647jpTXH8e7mM0iMi0C/VqGo0OhQozUg7UIxzhZUYOfZNQh59QAACKhJREFUIuw8U4gavQFjuzbFy2Pi0CTAW+ynQSKzqHECgMLCQvztb39DVVUVli5das2ciIhsivXLvXWJDsSaRwdg74Vi/LA/B78ez8fK9FzkXy4DAKR8ngYAaBLghel9YjGjXyzaRPiLmTI5EIkgCEJ9G6SmpiI1NbXOffPmzcPgwYMBADt27MDSpUvx9ddf19kmIyMDCoV4S89rNBp4e4v3l4GY8cWMPWvWLBgMBixbtkyU+ID7vvaOEF+lUqFnz56ixf8rS+sXIG4NE/t9dLf4OoOAy+VaPP632TAYBLzy3heICfREuK/M7pO/3e21d5TYwB3UL8EC+/btE0pLS01f9+nT56Zt0tPTLdm11Zw8edJt44sZe+jQoULv3r1Fiy8I7vvaO0J8sX/uzWFO/RIEcZ+L2O+ju8Z39/oldnyxn7u5P/MWTQ7fvHkz1qxZAwA4ffo0oqKiLNkNEZHdsX4RUWNYNMfp0Ucfxfz587FlyxbU1NTg1VdftXJaRES2wfpFRI1hUeMUEhKCzz//3Nq5EBHZHOsXETUGVw4nIiIiMhMbJyIiIiIzsXEiIiIiMhMbJyIiIiIzsXEiIiIiMhMbJyIiIiIzsXEiIiIiMhMbJyIiIiIzsXEiIiIiMhMbJyIiIiIzsXEiIiIiMhMbJyIiIiIzsXEiIiIiMhMbJyIiIiIzsXEiIiIiMhMbJyIiIiIzsXEiIiIiMhMbJyIiIiIzsXEiIiIiMhMbJyIiIiIzsXEiIiIiMhMbJyIiIiIzsXEiIiIiMlOjGqeioiL07t0b+/bts1Y+RER2wfpFRJZoVOP09ttvIyYmxlq5EBHZDesXEVnC4sZp79698PX1Rbt27ayZDxGRzbF+EZGlJIIgCHf6oJqaGsyePRuffPIJ3nrrLdx9993o27dvnW0yMjKsliQROY+ePXuKnUK9zKlfAGsYkTsyp37JG9ogNTUVqampde4bMmQIpk6dioCAgEYFJyKyJUvrF8AaRkS3ZtGIU0pKCgwGAwAgOzsbISEh+OCDD9C2bVurJ0hEZE2sX0TUGBY1TjeaP3/+bYe6iYgcGesXEd0pm6/jJNYpv8XFxXj44Ycxc+ZMpKSk4MiRI3aLrdPp8Pzzz2P69Om45557kJ6ebrfYtfbv34/+/ftj27Ztdo371ltvYdq0aUhJScHRo0ftGhsAzpw5g8TERHz//fd2j/32229j2rRpmDx5MjZv3mzX2Gq1Go8//jhmzJiBqVOn2v19BwCNRoPExESsXr3a7rFtxR3rFyB+DWP9sn/9AsSrYY5QvwDza1iDc5wasmDBgnq/L9Ypv+vXr8eECRMwbtw47N+/Hx988AG+/vpru8Ret24dfHx88MMPP+Ds2bN44YUXsGrVKrvEBoyHH7755hv06NHDbjEBY7G7dOkSVqxYgfPnz+PFF1/EihUr7BZfpVLh9ddfR//+/e0Ws1ZaWhrOnj2LFStWoKSkBHfffTfuuusuu8Xftm0bOnfujDlz5iAvLw+zZ8/GsGHD7BYfAD799FMEBgbaNWZjsX7dmpg1jPXL/vULELeGOUL9AsyvYY1unOoj5im/Dz74oOn2lStX0KRJE7vFHj9+PMaOHQsACAkJQWlpqd1iA0B4eDg++ugjvPTSS3aNu3fvXiQmJgIAWrdujbKyMlRWVsLPz88u8T09PfHFF1/giy++sEu8G/Xu3Rtdu3YFAAQEBECtVkOv10Mmk9kl/ujRo0237f15B4Dz58/j3LlzSEhIsGtcW3LX+gWIW8NYv+xfvwBxa5jY9Qu4sxpms0N1NTU1+Pjjj/Hkk0/aKkSDCgsLMXnyZHz66ad44okn7BbXw8MDXl5eAIClS5eaCpC9+Pj42O0X9o2KiooQHBxs+jokJASFhYV2iy+Xy+Ht7W23eDeSyWRQKBQAgFWrVmHIkCGivAcpKSl45pln8OKLL9o17sKFCzF//ny7xrQld65fgLg1jPVLHI5Qw8SqX8Cd1TCrjDg15pRfW8WfN28eBg8ejJ9++gk7duzACy+8YJOh7vpiL1++HCdOnMCSJUusHtec+GJr5HkHTmnr1q1YtWqVXQ+r3OjHH39EZmYmnn32Wfx/O3fMmjoURwH8WFq30kJQmvYTlC7F0Q5+CkE6dXLwA4gQqK7pJASKg4KgQsCldCjUggTcxEmi4NKhIhQJrSKUIhTeUCqv8Hwv8ry5lpzfFLMcp8Ofm3vv7e0tAoGA8Mybmxucnp7+2Fu4/dxf/8oX3WHsr80js8Nk9BeweoetZXCKx+OIx+Pf3n0d+a3Vanh6ekK32xV25PdP+e12G9PpFHt7e4jFYkin02vPXZYNfBZCs9nE9fU1dnZ2hGT/LV+GcDgMx3EWv8fjMUKhkMR/5K1Wq4VCoYBisYjd3V1Ps23bhqIoUFUVx8fH+Pj4wMvLCxRFEZ5tWRaGwyEsy8Lz8zOCwSAODg4QjUaFZ6+Dn/trWT7gTYexvzaLrA6T2V/A6h0mbI+TaZqL568jv17ek9JoNNDv93FxcYHBYABVVT3LHg6HME0T1Wp1sdztB2dnZzAMA4lEAr1eD+Fw2LP9AbLNZjNcXV2hXC5jf3/f8/xOp4PRaARN0+A4Dt7e3r59dhApn88vng3DwNHR0Y8Zmpbxc38B/uwwP/cXILfDZPYXsHqHCd0cLlMqlUImk8HDwwPm8zlyuZxn2fV6HZPJBMlkcvGuVCohGAx6km9ZFkqlEh4fH9Hr9VCpVDxZdo1EIjg5OUEikUAgEEA2mxWe+TvbtqHrOkajEba3t3F/fw/DMDwpgbu7O7y+vn7bi6LrOg4PD4VnA58rJJqm4fz8HO/v77i8vMTWlvDbRkgQmf0FyO0w9pf3/QXI7bCf1l//fQEmERERkV9s7khHREREtGE4OBERERG5xMGJiIiIyCUOTkREREQucXAiIiIicomDExEREZFLHJyIiIiIXOLgREREROTSL3iZV+YIXb1vAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 720x1080 with 2 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "qEYYEzmro-DS",
        "colab_type": "text"
      },
      "source": [
        "In this case, both $x_1=(1,0)$ and $x_2=(0,1)$ are eigenvectors of matrix $\\mathbf{A}$ with eigenvalues 1 and 3 respectively. Another thing you may already notice is: these two eigenvectors are along the major and minor axes of the eclipse (principal axes).\n",
        "\n",
        "An eclipse can be thought of as a circle stretched or shrunk along its principal axes. In this example matrix $\\mathbf{A}$ transforms the initial unit circle by stretching it along the direction of its eigenvectors: $x_1$ (minor axis) and $x_2$ (major axis), or $t_1$ and $t_2$ since they have the same direction with $x_1$ and $x_2$ respectively. This is a special property of symmetric matrix. \n",
        "\n",
        "- **Property 1:** A symmetric matrix transforms a vector by stretching or shrinking it along its eigenvectors by its corresponding eigenvalues. \n",
        "\n",
        "So when you have more stretching in the direction of an eigenvector, the eigenvalue corresponding to that eigenvector will be greater. That's why the eigenvector $x_2$ has a larger eigenvalue 3 than $x_1$.\n",
        "\n",
        "Recall a **symmetric matrix** is a square matrix with property:\n",
        "\n",
        "$$A_{i,j} = A_{j, i}$$\n",
        "\n",
        "In this example:\n",
        "\n",
        "$$A = \\begin{bmatrix} 1 & 0 \\\\ 0 & 3 \\end{bmatrix}$$\n",
        "\n",
        "Is a $2*2$ symmetric matrix (elements on the main diagonal do not matter).\n",
        "\n",
        "Another importatant property of symmetric metrices is: \n",
        "- **Property 2:** an $n×n$ symmetric matrix has $n$ linearly independent eigenvectors and are perpendicular to each other and form orthogonal vectors\n",
        "\n",
        "## Calculate eigenvalues and eigenvectors with Numpy\n",
        "\n",
        "Now Let's see the calculation of eigenvector and eigenvalues using Numpy and verify the examples above."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "t32wEYbYOmKs",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "from numpy import linalg as LA\n",
        "A = np.array([[1, 0],\n",
        "              [0, 3]]) \n",
        "w, v = LA.eig(A)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Qs5lCeQ0roI8",
        "colab_type": "text"
      },
      "source": [
        "- `w` contains the eigenvalues, \n",
        "- `v` contains the corresponding eigenvectors, one eigenvector per column"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "HS0r8pQTnrAO",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 72
        },
        "outputId": "23db9fad-79ab-4b50-e38f-161951585a26"
      },
      "source": [
        "print(w)\n",
        "print(v)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[1. 3.]\n",
            "[[1. 0.]\n",
            " [0. 1.]]\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "XBDJlLt-rU26",
        "colab_type": "text"
      },
      "source": [
        "The eigenvalue `w[0]` goes with the 0-th column of `v`. The eigenvalue `w[1]` goes with column 1, meaning the matrix: <br><br>\n",
        "\n",
        "$$A = \\begin{bmatrix} 1 & 0 \\\\ 0 & 3 \\end{bmatrix}$$\n",
        "\n",
        "has\n",
        "1. **eigenvector:** $\\begin{bmatrix} 1 \\\\0 \\end{bmatrix}$ and the corresponding **eigenvalue:** 1\n",
        "2. **eigenvector:** $\\begin{bmatrix} 0 \\\\1 \\end{bmatrix}$ and the corresponding **eigenvalue:** 3\n",
        "\n",
        "This is consistent with our analysis above! **Note the eigenvectors are normalized so their l2 norms will always be 1.**"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "1I8zLps1ioC6",
        "colab_type": "text"
      },
      "source": [
        "## Eigendecomposition for Symmetric Matrix\n",
        "\n",
        "**Matrix decomposition** is a useful tool for reducing a matrix to their constituent parts in order to simplify a range of more complicated operations. Perhaps the most used type of matrix decomposition is the **eigendecomposition** that decomposes a matrix into eigenvectors and eigenvalues.\n",
        "\n",
        "A $\\color{red}{\\text{symmetric matrix}}$ is orthogonally diagonalizable. It means for a $n×n$ symmetric matrix $\\mathbf{A}$ with $n$ linearly independent eigenvectors $v_i$ and corresponding eigenvalues $\\lambda_i$ , we can decompose it as:\n",
        "\n",
        "<br>$$A = Q\\Lambda Q^{-1}$$<br>\n",
        "\n",
        "where $Q$ is the square $n × n$ matrix whose i-th column is the eigenvector $v_i$ of A, since $Q$ is formed from the eigenvectors of $A$,  it is guaranteed to be an **orthogonal matrix**, therefore $Q^{-1} = Q^T$. $\\Lambda$ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, $\\Lambda_{ii} = λ_i$. That is:\n",
        "\n",
        "$$A = \\begin{bmatrix} v_1 & v_2 & ... & v_n \\end{bmatrix} \\begin{bmatrix} \n",
        "\\lambda_1 & 0 & ... & 0\\\\\n",
        "0 & \\lambda_2 & ... & 0\\\\\n",
        "\\vdots & \\vdots & ... & \\vdots\\\\\n",
        "0 & 0 & ... & \\lambda_n\n",
        "\\end{bmatrix} \\begin{bmatrix} v_1^T \\\\ v_2^T \\\\ \\vdots \\\\ v_n^T \\end{bmatrix}$$\n",
        "\n",
        "Let's go one step further:\n",
        "\n",
        "$$A = \\begin{bmatrix} \\lambda_1v_1 & \\lambda_2v_2 & ... & \\lambda_nv_n \\end{bmatrix} \\begin{bmatrix} v_1^T \\\\ v_2^T \\\\ \\vdots \\\\ v_n^T \\end{bmatrix}\n",
        "=\\lambda_1v_1v_1^T + \\lambda_2v_2v_2^T + ... + \\lambda_nv_nv_n^T$$\n",
        "\n",
        "So when we apply this matrix $A$ to some vector $x$:\n",
        "\n",
        "<br> $$Ax = \\lambda_1v_1v_1^Tx + \\lambda_2v_2v_2^Tx + ... + \\lambda_nv_nv_n^Tx$$ <br>\n",
        "\n",
        "\n",
        "Let's illustrate this concept with the earlier example:\n",
        "\n",
        "$$A = \\begin{bmatrix} 1 & 0 \\\\ 0 & 3 \\end{bmatrix}$$\n",
        "\n",
        "**eigenvector:** $\\begin{bmatrix} 1 \\\\0 \\end{bmatrix}$ and the corresponding **eigenvalue:** 1 <br>\n",
        "**eigenvector:** $\\begin{bmatrix} 0 \\\\1 \\end{bmatrix}$ and the corresponding **eigenvalue:** 3\n",
        "\n",
        "Thus, it can be decomposed as:\n",
        "\n",
        "$$A = \\begin{bmatrix} 1 & 0 \\\\ 0 & 3 \\end{bmatrix} = \n",
        "\\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} \\begin{bmatrix} \n",
        "1 & 0 \\\\\n",
        "0 & 3\n",
        "\\end{bmatrix} \\begin{bmatrix} 1 & 0 \\\\ 0 & 3 \\end{bmatrix} \n",
        "$$\n",
        "\n",
        "let's verify this with code:\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "xMmKbh_skg3e",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 108
        },
        "outputId": "46575f2c-8f87-49fe-a1e6-1f1379412d48"
      },
      "source": [
        "Q = np.array([[1, 0], [0, 1]])\n",
        "lamda = np.array([[1, 0], [0, 3]])\n",
        "Q_transpose = Q.T\n",
        "Q_inverse = np.linalg.inv(Q)\n",
        "print(np.matmul(np.matmul(Q, lamda), Q_transpose), '\\n')\n",
        "print(np.matmul(np.matmul(Q, lamda), Q_inverse))"
      ],
      "execution_count": 21,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[[1 0]\n",
            " [0 3]] \n",
            "\n",
            "[[1. 0.]\n",
            " [0. 3.]]\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Klox3KBoMc4m",
        "colab_type": "text"
      },
      "source": [
        "Great! now let's dig a little deeper:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "cxqHq6ZREnHQ",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "A = np.array([[1, 0],\n",
        "              [0, 3]]) \n",
        "v1 = np.array([[1], [0]])\n",
        "v2 = np.array([[0], [1]])\n",
        "lamda1 = 1\n",
        "lamda2 = 3"
      ],
      "execution_count": 87,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "LFI12fteNxpO",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 54
        },
        "outputId": "c5b6b21b-b530-4931-b9a0-9f192758c588"
      },
      "source": [
        "# Apply transformation A to a random vector x\n",
        "x = np.array([[2], [1]])\n",
        "Ax = np.matmul(A, x)\n",
        "print(Ax)"
      ],
      "execution_count": 88,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[[2]\n",
            " [3]]\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "AQRsCGKTMM7R",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 54
        },
        "outputId": "cee78147-689c-4ffc-b9e4-fe93d9cdfe4f"
      },
      "source": [
        "# Another expression\n",
        "vvt1x = np.matmul(np.matmul(v1, v1.T), x)\n",
        "vvt2x = np.matmul(np.matmul(v2, v2.T), x)\n",
        "print(lamda1*vvt1x + lamda2*vvt2x)"
      ],
      "execution_count": 89,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[[2]\n",
            " [3]]\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_LfJcQsBPRGX",
        "colab_type": "text"
      },
      "source": [
        "It works as expect! Actually, given $v$ is a unit vector, multiplying $vv^T$ by $x$ will give the **orthogonal projection of $x$ onto $v$**, so each of the matrices $v_iv_i^T$ is called the **projection matrix**. You can skip the code in the next cell and go straight to its visualization."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "Yq9gW3yzIwRi",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 481
        },
        "outputId": "39d3c86b-44d1-4ae8-82e1-16f0d2c10007"
      },
      "source": [
        "origin = [0], [0]   # origin point\n",
        "fig = plt.figure(figsize=(13, 7)) \n",
        "\n",
        "ax1 = fig.add_subplot(1, 2, 1)  \n",
        "ax1.quiver(*origin, x[0, 0], x[1, 0], \n",
        "          color='steelblue', scale_units='xy', \n",
        "          width=0.01, angles='xy', scale=1)\n",
        "ax1.quiver(*origin, v1[0, 0], v1[1, 0], linestyle='dashed',\n",
        "          color='purple', scale_units='xy', linewidth=1,\n",
        "          width=0.01, angles='xy', scale=1)\n",
        "ax1.quiver(*origin, vvt1x[0, 0], vvt1x[1, 0], \n",
        "          color='red', scale_units='xy', \n",
        "          width=0.01, angles='xy', scale=1)\n",
        "ax1.quiver(*origin, v2[0, 0], v2[1, 0], \n",
        "          color='purple', scale_units='xy', \n",
        "          width=0.01, angles='xy', scale=1)\n",
        "ax1.quiver(*origin, vvt2x[0, 0], vvt2x[1, 0],\n",
        "          color='red', scale_units='xy', \n",
        "          width=0.01, angles='xy', scale=1)\n",
        "\n",
        "ax1.plot([vvt1x[0, 0], x[0, 0]], [vvt1x[1, 0], x[1, 0]],\n",
        "         linestyle=':', color='red')\n",
        "ax1.plot([vvt2x[0, 0], x[0, 0]], [vvt2x[1, 0], x[1, 0]],\n",
        "         linestyle=':', color='red')\n",
        "ax1.text(x[0, 0]+0.1, x[1, 0]+0.1, \"$\\mathbf{x}$\",\n",
        "         color='steelblue', fontsize=14)\n",
        "ax1.text(v1[0, 0], v1[1, 0]+0.1, \"$\\mathbf{v_1}$\",\n",
        "         color='purple', fontsize=14)\n",
        "ax1.text(vvt1x[0, 0], vvt1x[1, 0]+0.1,\n",
        "         \"$\\mathbf{v_1v_1^Tx}$\", color='red', fontsize=14)\n",
        "ax1.text(v2[0, 0]+0.1, v2[1, 0]+0.1, \n",
        "         \"$\\mathbf{v_2}$\", color='purple', fontsize=14)\n",
        "ax1.text(vvt2x[0, 0]-1.1, vvt2x[1, 0]+0.1, \n",
        "         \"$\\mathbf{v_2v_2^Tx}$\", color='red', fontsize=14)\n",
        "\n",
        "ax1.set_xlim([-3, 3])\n",
        "ax1.set_ylim([-3, 3])\n",
        "ax1.set_aspect('equal')\n",
        "ax1.axhline(y=0, color='k')\n",
        "ax1.axvline(x=0, color='k')\n",
        "\n",
        "ax2 = fig.add_subplot(1, 2, 2)  \n",
        "ax2.quiver(*origin, x[0, 0], x[1, 0], \n",
        "          color='steelblue', scale_units='xy', \n",
        "          width=0.01, angles='xy', scale=1)\n",
        "ax2.quiver(*origin, vvt1x[0, 0], vvt1x[1, 0],\n",
        "          color='red', scale_units='xy', \n",
        "          width=0.01, angles='xy', scale=1)\n",
        "ax2.quiver(*origin, vvt2x[0, 0], vvt2x[1, 0], \n",
        "          color='red', scale_units='xy', \n",
        "          width=0.01, angles='xy', scale=1)\n",
        "ax2.quiver(*origin, lamda1*vvt1x[0, 0], lamda1*vvt1x[1, 0], \n",
        "          color='green', scale_units='xy', \n",
        "          width=0.01, angles='xy', scale=1)\n",
        "ax2.quiver(*origin, lamda2*vvt2x[0, 0], lamda2*vvt2x[1, 0], \n",
        "          color='green', scale_units='xy', \n",
        "          width=0.01, angles='xy', scale=1)\n",
        "ax2.quiver(*origin, lamda1*vvt1x[0, 0]+lamda2*vvt2x[0, 0], \n",
        "           lamda1*vvt1x[1, 0]+lamda2*vvt2x[1, 0], \n",
        "           color='orange', scale_units='xy', \n",
        "           width=0.01, angles='xy', scale=1)\n",
        "ax2.plot([lamda1*vvt1x[0, 0], lamda1*vvt1x[0, 0]+lamda2*vvt2x[0, 0]],\n",
        "         [lamda1*vvt1x[1, 0], lamda1*vvt1x[1, 0]+lamda2*vvt2x[1, 0]], \n",
        "         linestyle=':', color='green')\n",
        "ax2.plot([lamda2*vvt2x[0, 0], lamda1*vvt1x[0, 0]+lamda2*vvt2x[0, 0]],\n",
        "         [lamda2*vvt2x[1, 0], lamda1*vvt1x[1, 0]+lamda2*vvt2x[1, 0]],\n",
        "         linestyle=':', color='green')\n",
        "\n",
        "ax2.text(x[0, 0]+0.1, x[1, 0]+0.1, \n",
        "         \"$\\mathbf{x}$\", color='steelblue', fontsize=14)\n",
        "ax2.text(vvt1x[0, 0], vvt1x[1, 0]+0.3, \n",
        "         \"$\\mathbf{v_1v_1^Tx}$\", color='red', fontsize=14)\n",
        "ax2.text(vvt2x[0, 0]-1.2, vvt2x[1, 0], \n",
        "         \"$\\mathbf{v_2v_2^Tx}$\", color='red', fontsize=14)\n",
        "ax2.text(lamda1*vvt1x[0, 0], lamda1*vvt1x[1, 0]-0.5, \n",
        "         \"$\\mathbf{\\lambda_1v_1v_1^Tx}$\", color='green', fontsize=14)\n",
        "ax2.text(lamda2*vvt2x[0, 0]-1.6, lamda2*vvt2x[1, 0], \n",
        "         \"$\\mathbf{\\lambda_2v_2v_2^Tx}$\", color='green', fontsize=14)\n",
        "ax2.text(lamda1*vvt1x[0, 0]+lamda2*vvt2x[0, 0], \n",
        "         lamda1*vvt1x[1, 0]+lamda2*vvt2x[1, 0], \n",
        "         \"$\\mathbf{Ax=\\lambda_1v_1v_1^Tx+\\lambda_2v_2v_2^Tx}$\", \n",
        "         color='orange', fontsize=14)\n",
        "\n",
        "ax2.set_xlim([-4, 4])\n",
        "ax2.set_ylim([-4, 4])\n",
        "ax2.set_aspect('equal')\n",
        "ax2.axhline(y=0, color='k')\n",
        "ax2.axvline(x=0, color='k')\n",
        "fig.suptitle('Projection of $x$ onto $v_1$ and $v_2$', fontsize=17)\n",
        "plt.tight_layout()\n",
        "plt.show()"
      ],
      "execution_count": 152,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 936x504 with 2 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "4IgjHZCbaOoT",
        "colab_type": "text"
      },
      "source": [
        "The eigenvalues play an important role here since they can be thought of as a multiplier. The projection matrix projects $x$ onto each $v_i$,and scales the length of the vector projection by factor eigenvalue. The bigger the eigenvalue, the bigger the length of the resulting vector. Assume all the eigenvalues $\\lambda_i$ have been sorted in descending order:\n",
        "\n",
        "<br> $$A = \\lambda_1v_1v_1^T + \\lambda_2v_2v_2^T + ... + \\lambda_nv_nv_n^T$$ <br>\n",
        "\n",
        "with\n",
        "\n",
        "$$\\lambda_1 \\geq \\lambda_2 \\geq ...  \\geq \\lambda_n$$<br>\n",
        "\n",
        "We can simply take the top $k$ projection vectors to aprroximate the original symmetric matrix $A$:\n",
        "\n",
        "<br> $$A \\approx \\lambda_1v_1v_1^T + \\lambda_2v_2v_2^T + ... + \\lambda_kv_kv_k^T$$ <br>\n",
        "\n",
        "If in the original matrix $A$, the remaining $(n-k)$ eigenvalues that we leave out are very small and close to zero, then the approximated matrix is very similar to the original matrix, and we have a good approximation. This method is widely used to map high-dimensional data into a lower dimension space. However, it only works for symmetric matrices. \n",
        "\n",
        "Now you have learned about eigendecomposition. In the next note I will introduce you the **Singular Value Decomposition** (SVD) which will work for non-symmetic matrix."
      ]
    }
  ]
}