ep28_Eigenvectors.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "ep28_Eigenvectors.ipynb",
      "provenance": [],
      "collapsed_sections": []
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "# Episode 28. What Are Eigenvalues and Eigenvectors"
      ],
      "metadata": {
        "id": "73cP11F5S0Zo"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Import Libraries"
      ],
      "metadata": {
        "id": "dUMuX5LXS_yW"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "from numpy import linalg as LA\n",
        "import matplotlib.pyplot as plt"
      ],
      "metadata": {
        "id": "oe5PrdNuS4pt"
      },
      "execution_count": 9,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Set Matrix A"
      ],
      "metadata": {
        "id": "HisfzGKXUKKp"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "A = np.array([[1,2],[3,-4]])\n",
        "print(A)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "qdXIEmQLTCm9",
        "outputId": "35d49d97-d0a9-4444-90ee-7e3374162bc7"
      },
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[[ 1  2]\n",
            " [ 3 -4]]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Get Eigenvalues and Eigenvectors"
      ],
      "metadata": {
        "id": "17ooco7-VIb5"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# use numpy to get eigenvalues (w) and eigenvectors (v)\n",
        "w, v = LA.eig(A)\n",
        "print(\"w=\",w)\n",
        "print(\"v=\",v)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "QoTObbnPTDA9",
        "outputId": "4213fd27-eeb2-4b3c-d544-afa4d5c70f67"
      },
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "w= [ 2. -5.]\n",
            "v= [[ 0.89442719 -0.31622777]\n",
            " [ 0.4472136   0.9486833 ]]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# sort by largest eigenvalue\n",
        "idx = w.argsort()[::-1]   \n",
        "w = w[idx]\n",
        "v = v[:,idx]\n",
        "print(\"w=\",w)\n",
        "print(\"v=\",v)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "dqueKkiITDFq",
        "outputId": "ccb10910-3ad6-4408-b76a-1fc14cbc152e"
      },
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "w= [ 2. -5.]\n",
            "v= [[ 0.89442719 -0.31622777]\n",
            " [ 0.4472136   0.9486833 ]]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# v1 & w1\n",
        "w1 = w[0]\n",
        "v1 = v[:,0]/np.min(np.abs(v[:,0]))\n",
        "print(\"w1=\",w1,\"; v1=\",v1)\n",
        "\n",
        "# v2 & w2\n",
        "w2 = w[1]\n",
        "v2 = v[:,1]/np.min(np.abs(v[:,1]))\n",
        "print(\"w2=\",w2,\"; v2=\",v2)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "JyHMrZcgWJmJ",
        "outputId": "225e3420-28fb-48ee-ddab-3d303181e090"
      },
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "w1= 2.0 ; v1= [2. 1.]\n",
            "w2= -5.0 ; v2= [-1.  3.]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Check whether A*v = w*v"
      ],
      "metadata": {
        "id": "pcWOSsH7b0sS"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Check A*v1 = lambda1*v1\n",
        "A_v1 = np.matmul(A,v1)\n",
        "w_v1= w1*v1\n",
        "print(\"A*v1=\",A_v1)\n",
        "print(\"w1*v1=\",w_v1)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "dPNtYl8QTDMM",
        "outputId": "0609b4dc-c17e-43ea-f31a-8699f2aa4ad6"
      },
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "A*v1= [4. 2.]\n",
            "w1*v1= [4. 2.]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Check A*v2 = lambda1*v2\n",
        "A_v2 = np.matmul(A,v2)\n",
        "w_v2= w2*v2\n",
        "print(\"A*v2=\",A_v2)\n",
        "print(\"w2*v2=\",w_v2)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "7ptHLHtwXVSK",
        "outputId": "afc901c7-0ab6-45d2-a4e2-fdab7be2ee4d"
      },
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "A*v2= [  5. -15.]\n",
            "w2*v2= [  5. -15.]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Geometric Interpretation"
      ],
      "metadata": {
        "id": "yNpMuu8i2MxA"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Pick an arbitrary vector"
      ],
      "metadata": {
        "id": "XKF5gcIv0i3d"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# pick arbitrary vector\n",
        "vp = np.array([1,1])\n",
        "# origin vector\n",
        "vo = np.array([[0,0],[0,0]])\n",
        "\n",
        "fig, ax = plt.subplots()\n",
        "q = ax.quiver(vo[0],vo[1],vp[0],vp[1],units='xy',scale=1)\n",
        "plt.title('Vector vp',fontsize=10)\n",
        "ax.set_aspect('equal')\n",
        "plt.xlim(-1,2)\n",
        "plt.ylim(-1,2)\n",
        "plt.grid()\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 279
        },
        "id": "3gdL7NhneK6Q",
        "outputId": "c21077b5-90f8-426b-e1e5-6438b7ffa652"
      },
      "execution_count": 10,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": "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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Multiplying a matrix to a vector does two things: ***rescale*** & ***rotate*** the vector"
      ],
      "metadata": {
        "id": "_kNzVOHU0ocq"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# pick arbitrary vector\n",
        "vp = np.array([1,1])\n",
        "# origin vector\n",
        "vo = np.array([[0,0],[0,0]])\n",
        "# v2 & A*v2\n",
        "A_vp = np.matmul(A,vp)\n",
        "v_Av = np.vstack((vp,A_vp))\n",
        "print(\"vp=\",vp)\n",
        "print(\"A*vp=\",A_vp)\n",
        "print(\"v_Av=\",v_Av)\n",
        "\n",
        "fig, ax = plt.subplots()\n",
        "q = ax.quiver(vo[:,0],vo[:,1],v_Av[:,0],v_Av[:,1],units='xy',scale=1,color=['k','r'])\n",
        "plt.title('vp vs. A*vp',fontsize=10)\n",
        "ax.set_aspect('equal')\n",
        "plt.xlim(0,3)\n",
        "plt.ylim(-1,2)\n",
        "plt.grid()\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 348
        },
        "id": "8igynyI_2QHm",
        "outputId": "85a28183-604c-46eb-c6f5-045a3d106a8c"
      },
      "execution_count": 11,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "vp= [1 1]\n",
            "A*vp= [ 3 -1]\n",
            "v_Av= [[ 1  1]\n",
            " [ 3 -1]]\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": "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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Compare eigenvector v and A*v"
      ],
      "metadata": {
        "id": "FUh7UtsgWTJf"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### (1) v1 vs. A*v1"
      ],
      "metadata": {
        "id": "B7gNoer4afSg"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# origin vector\n",
        "vo = np.array([[0,0],[0,0]])\n",
        "# v1 & A*v1\n",
        "A_v1 = np.matmul(A,v1)\n",
        "vv = np.vstack((v1,A_v1))\n",
        "print(\"v1=\",v1)\n",
        "print(\"A*v1=\",A_v1)\n",
        "print(\"vv=\",vv)\n",
        "\n",
        "fig, ax = plt.subplots()\n",
        "q = ax.quiver(vo[:,0],vo[:,1],vv[:,0],vv[:,1],units='xy',scale=1,color=['k','r'])\n",
        "plt.title('v1 vs. A*v1',fontsize=10)\n",
        "ax.set_aspect('equal')\n",
        "plt.xlim(-1,3)\n",
        "plt.ylim(-1,3)\n",
        "plt.grid()\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 348
        },
        "id": "5MtRYZHADRT9",
        "outputId": "4589deac-925e-4ff9-d757-b3e9c912d425"
      },
      "execution_count": 12,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "v1= [2. 1.]\n",
            "A*v1= [4. 2.]\n",
            "vv= [[2. 1.]\n",
            " [4. 2.]]\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": "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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### (2) v2 vs. A*v2"
      ],
      "metadata": {
        "id": "ln6hinXYalh4"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# origin vector\n",
        "vo = np.array([[0,0],[0,0]])\n",
        "# v2 & A*v2\n",
        "A_v2 = np.matmul(A,v2)\n",
        "vvv = np.vstack((v2,A_v2))\n",
        "print(\"v2=\",v2)\n",
        "print(\"A*v2=\",A_v2)\n",
        "print(\"vvv=\",vvv)\n",
        "\n",
        "fig, ax = plt.subplots()\n",
        "q = ax.quiver(vo[:,0],vo[:,1],vvv[:,0],vvv[:,1],units='xy',scale=1,color=['k','r'])\n",
        "plt.title('v2 vs. A*v2',fontsize=10)\n",
        "ax.set_aspect('equal')\n",
        "plt.xlim(-10,10)\n",
        "plt.ylim(-15,5)\n",
        "plt.grid()\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 348
        },
        "id": "YFe6pCrqW1J-",
        "outputId": "65000f19-6f9d-4dd3-e87d-8e1ce4f6019a"
      },
      "execution_count": 13,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "v2= [-1.  3.]\n",
            "A*v2= [  5. -15.]\n",
            "vvv= [[ -1.   3.]\n",
            " [  5. -15.]]\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": "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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    }
  ]
}