├── .DS_Store
├── .ipynb_checkpoints
    ├── image_compression-checkpoint.ipynb
    ├── pi_matrix_decoposition-checkpoint.ipynb
    ├── plotting_ellipses-checkpoint.ipynb
    └── svd-checkpoint.ipynb
├── README.md
├── image_compression.ipynb
├── pi_matrix_decoposition.ipynb
└── plotting_ellipses.ipynb


/.DS_Store:
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https://raw.githubusercontent.com/luisguiserrano/singular_value_decomposition/20163a9ef0939823c61cf381bc1621f52fc7f698/.DS_Store


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/README.md:
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 1 | # singular_value_decomposition
 2 | 
 3 | This is the repo for the video "Singular value decomposition and applications to image compression".
 4 | https://youtu.be/DG7YTlGnCEo
 5 | 
 6 | Notebooks:
 7 | 
 8 | - image_compression.ipynb: The image compression example in the video.
 9 | - pi_matrix.ipynb: The random matrix example in the video.
10 | - plotting_ellipses.ipynb: Some rotation and translation plots.


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/pi_matrix_decoposition.ipynb:
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  1 | {
  2 |  "cells": [
  3 |   {
  4 |    "cell_type": "markdown",
  5 |    "metadata": {},
  6 |    "source": [
  7 |     "# Example with random matrix"
  8 |    ]
  9 |   },
 10 |   {
 11 |    "cell_type": "code",
 12 |    "execution_count": 13,
 13 |    "metadata": {},
 14 |    "outputs": [],
 15 |    "source": [
 16 |     "import numpy as np\n",
 17 |     "from numpy.linalg import svd\n",
 18 |     "import matplotlib.pyplot as plt\n",
 19 |     "from matplotlib.pyplot import imshow"
 20 |    ]
 21 |   },
 22 |   {
 23 |    "cell_type": "code",
 24 |    "execution_count": 14,
 25 |    "metadata": {},
 26 |    "outputs": [],
 27 |    "source": [
 28 |     "vmin = 0\n",
 29 |     "vmax = 1\n",
 30 |     "image_bias = 1 # sometimes 1\n",
 31 |     "def plot_svd(A):\n",
 32 |     "    n = len(A)\n",
 33 |     "    U, S, V = svd(A)\n",
 34 |     "\n",
 35 |     "    imgs = []\n",
 36 |     "    for i in range(n):\n",
 37 |     "        imgs.append(S[i]*np.outer(U[:,i],V[i]))\n",
 38 |     "\n",
 39 |     "    combined_imgs = []\n",
 40 |     "    for i in range(n):\n",
 41 |     "        img = sum(imgs[:i+1])\n",
 42 |     "        combined_imgs.append(img)\n",
 43 |     "        print(np.round(img,2))\n",
 44 |     "        print(\"=====\")\n",
 45 |     "    \n",
 46 |     "    #print(imgs)\n",
 47 |     "    \n",
 48 |     "    print(combined_imgs)\n",
 49 |     "    \n",
 50 |     "    return U,S,V"
 51 |    ]
 52 |   },
 53 |   {
 54 |    "cell_type": "code",
 55 |    "execution_count": 15,
 56 |    "metadata": {},
 57 |    "outputs": [
 58 |     {
 59 |      "name": "stdout",
 60 |      "output_type": "stream",
 61 |      "text": [
 62 |       "[[2.51 2.37 2.22 1.97]\n",
 63 |       " [6.07 5.72 5.37 4.77]\n",
 64 |       " [5.63 5.31 4.99 4.43]\n",
 65 |       " [7.88 7.43 6.98 6.19]]\n",
 66 |       "=====\n",
 67 |       "[[3.15 1.41 3.79 0.56]\n",
 68 |       " [4.87 7.53 2.44 7.43]\n",
 69 |       " [5.28 5.85 4.12 5.22]\n",
 70 |       " [8.85 5.96 9.36 4.03]]\n",
 71 |       "=====\n",
 72 |       "[[3.1  0.96 3.93 0.99]\n",
 73 |       " [5.03 8.99 1.98 6.  ]\n",
 74 |       " [4.98 3.01 5.01 8.  ]\n",
 75 |       " [8.96 7.02 9.03 3.  ]]\n",
 76 |       "=====\n",
 77 |       "[[3. 1. 4. 1.]\n",
 78 |       " [5. 9. 2. 6.]\n",
 79 |       " [5. 3. 5. 8.]\n",
 80 |       " [9. 7. 9. 3.]]\n",
 81 |       "=====\n",
 82 |       "[array([[2.5129268 , 2.37029143, 2.22496161, 1.97491746],\n",
 83 |       "       [6.0691682 , 5.72467824, 5.37368069, 4.7697793 ],\n",
 84 |       "       [5.63425954, 5.31445527, 4.98860976, 4.42798315],\n",
 85 |       "       [7.87808152, 7.43091645, 6.97530422, 6.19141025]]), array([[3.15001871, 1.40791218, 3.78880714, 0.55746973],\n",
 86 |       "       [4.8743301 , 7.5295786 , 2.44075653, 7.42814078],\n",
 87 |       "       [5.27953502, 5.85029558, 4.11788081, 5.2171997 ],\n",
 88 |       "       [8.84904357, 5.96419913, 9.35868823, 4.03114429]]), array([[3.10337436, 0.96150448, 3.92917454, 0.99445918],\n",
 89 |       "       [5.02692418, 8.98997372, 1.98155328, 5.99855688],\n",
 90 |       "       [4.98239762, 3.00655494, 5.01206002, 8.00094348],\n",
 91 |       "       [8.95887283, 7.01531532, 9.02817769, 3.0022044 ]]), array([[3., 1., 4., 1.],\n",
 92 |       "       [5., 9., 2., 6.],\n",
 93 |       "       [5., 3., 5., 8.],\n",
 94 |       "       [9., 7., 9., 3.]])]\n"
 95 |      ]
 96 |     }
 97 |    ],
 98 |    "source": [
 99 |     "P = np.array([[3,1,4,1],[5,9,2,6],[5,3,5,8],[9,7,9,3]])\n",
100 |     "U,S,V = plot_svd(P)"
101 |    ]
102 |   },
103 |   {
104 |    "cell_type": "code",
105 |    "execution_count": null,
106 |    "metadata": {},
107 |    "outputs": [],
108 |    "source": []
109 |   }
110 |  ],
111 |  "metadata": {
112 |   "kernelspec": {
113 |    "display_name": "Python 3",
114 |    "language": "python",
115 |    "name": "python3"
116 |   },
117 |   "language_info": {
118 |    "codemirror_mode": {
119 |     "name": "ipython",
120 |     "version": 3
121 |    },
122 |    "file_extension": ".py",
123 |    "mimetype": "text/x-python",
124 |    "name": "python",
125 |    "nbconvert_exporter": "python",
126 |    "pygments_lexer": "ipython3",
127 |    "version": "3.7.6"
128 |   }
129 |  },
130 |  "nbformat": 4,
131 |  "nbformat_minor": 4
132 | }
133 | 


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/plotting_ellipses.ipynb:
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  1 | {
  2 |  "cells": [
  3 |   {
  4 |    "cell_type": "code",
  5 |    "execution_count": 5,
  6 |    "metadata": {},
  7 |    "outputs": [],
  8 |    "source": [
  9 |     "import numpy as np\n",
 10 |     "from numpy.linalg import svd\n",
 11 |     "import matplotlib.pyplot as plt\n",
 12 |     "from matplotlib.pyplot import imshow"
 13 |    ]
 14 |   },
 15 |   {
 16 |    "cell_type": "code",
 17 |    "execution_count": 6,
 18 |    "metadata": {},
 19 |    "outputs": [],
 20 |    "source": [
 21 |     "# SVD\n",
 22 |     "\n",
 23 |     "def plot(A, n=100):\n",
 24 |     "    x = np.array([[np.cos(2*np.pi*i/n), np.sin(2*np.pi*i/n)] for i in range(n)])\n",
 25 |     "    y = np.dot(A, x.T).T\n",
 26 |     "    for i in range(n):\n",
 27 |     "        plt.scatter(y[i,0], y[i,1], color=[(n-i)/n, 0, i/n])\n",
 28 |     "\n",
 29 |     "def plot_circles(A, xlim=4, ylim=4, n=100):\n",
 30 |     "    U,S,V = svd(A)\n",
 31 |     "    sigma = np.diag(S)\n",
 32 |     "\n",
 33 |     "    I = np.identity(len(A))\n",
 34 |     "    plt.xlim(-xlim, xlim)\n",
 35 |     "    plt.ylim(-ylim, ylim)\n",
 36 |     "    plot(I,n)\n",
 37 |     "    plt.title(\"Unit circle\")\n",
 38 |     "    plt.show()\n",
 39 |     "    plt.xlim(-xlim, xlim)\n",
 40 |     "    plt.ylim(-ylim, ylim)\n",
 41 |     "    plot(V,n)\n",
 42 |     "    print(\"Rotating\", np.arccos(V[0][0]), 'or', np.arccos(V[0][0])*180/np.pi, \"degrees\")\n",
 43 |     "    print(np.round(V,2))\n",
 44 |     "    plt.title(\"Vt\")\n",
 45 |     "    plt.show()\n",
 46 |     "    plt.xlim(-xlim, xlim)\n",
 47 |     "    plt.ylim(-ylim, ylim)\n",
 48 |     "    plot(np.dot(sigma,V),n)\n",
 49 |     "    plt.title(\"sigma Vt\")\n",
 50 |     "    print(np.round(sigma,2))\n",
 51 |     "    print(\"Stretching x-axis by\", sigma[0][0])\n",
 52 |     "    print(\"Stretching y-axis by\", sigma[1][1])\n",
 53 |     "    plt.show()\n",
 54 |     "    plt.xlim(-xlim, xlim)\n",
 55 |     "    plt.ylim(-ylim, ylim)\n",
 56 |     "    plot(np.dot(np.dot(U,sigma),V),n)\n",
 57 |     "    print(\"Rotating\", -np.arccos(U[0][0]), 'or', -np.arccos(U[0][0])*180/np.pi, \"degrees\")\n",
 58 |     "    plt.title(\"U sigma Vt\")\n",
 59 |     "    print(np.round(U,2))\n",
 60 |     "    plt.show()\n",
 61 |     "    plt.xlim(-xlim, xlim)\n",
 62 |     "    plt.ylim(-ylim, ylim)\n",
 63 |     "    plot(A,n)\n",
 64 |     "    print(\"Original transformation\")\n",
 65 |     "    plt.title(\"A\")\n",
 66 |     "    print(np.round(A,2))\n",
 67 |     "    \n",
 68 |     "    return U, S, V"
 69 |    ]
 70 |   },
 71 |   {
 72 |    "cell_type": "code",
 73 |    "execution_count": 7,
 74 |    "metadata": {},
 75 |    "outputs": [
 76 |     {
 77 |      "data": {
 78 |       "image/png": 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F2zcdGhCTpo/hrufr88FNZvUwLD5Pw+yd7qrOmYxpPfhXaUzrCK7qnNmkisyK4/s0zGo0d8EkIDu3sWPzPo6bNoarOme+3W72TuLQMKuDuQsmOSTsXcGHp8zMLJlDw8zMkjk0zMwsmUPDzMySOTTMzCyZQ8PMzJI5NMzMLJlDw8zMkhVyc5+kvwMuBN4ANgBXRMQrFfo9D/wGOADsr/WZKGZmVqyi9jSWAb8bEf8D+E/gy/30/WhEnOnAMDMb+goJjYh4MCL25y8fA6YWsR4zM2usRpzTuBK4v8q8AB6UtFJSe38LkdQuqUdSz86dO+tepJmZHd6gz2lIWg4cX2FWR0T8KO/TAewHuqss5pyI2CrpOGCZpGcj4uFKHSOiC+iC7PM0Blu3mZkN3qBDIyLm9jdf0mXAJ4E5UeWTniJia/59h6R7gdlAxdAwM7PmK+TwlKTzgS8Bn4qI3ip9xko6qm8aOA9YW0Q9ZmZWH0Wd07gVOIrskNNqSYsAJE2WtDTvMwl4RNKTwC+Bn0TETwuqx8zM6qCQ+zQi4r1V2rcC8/LpjcAZRazfzMyK4TvCzcwsmUPDzMySOTTMzCyZQ8PMzJI5NMzMLJlDw8zMkjk0zMwsmUPDzMySOTTMzCyZQ8PMzJI5NMzMLJlDw8zMkjk0zMwsmUPDzMySOTTMzCyZQ8PMzJI5NMzMLJlDw8zMkhUWGpJulPRi/hnhqyXNq9LvfEnrJK2XdF1R9ZiZWe0K+YzwEt+MiIXVZkpqAb4FfBzYAjwhaUlE/KrguszMbBCafXhqNrA+IjZGxBvAXcBFTa7JzMyqKDo0vijpKUm3SxpfYf4U4IWS11vytkNIapfUI6ln586dRdRqZmaHUVNoSFouaW2Fr4uA7wAnA2cC24CvV1pEhbaotK6I6IqItohomzhxYi1lm5nZINV0TiMi5qb0k/Rd4McVZm0BTix5PRXYWktNZmZWnCKvnjqh5OWngbUVuj0BnCLpJEmjgfnAkqJqMjOz2hR59dTXJJ1JdrjpeeDPACRNBm6LiHkRsV/SF4EHgBbg9oh4usCazMysBoWFRkR8tkr7VmBeyeulwNKi6jAzs/pp9iW3ZmY2jDg0zMwsmUPDzMySOTTMzCyZQ8PMzJI5NMzMLJlDw8zMkjk0zMwsmUPDzMySOTTMzCyZQ8PMzJI5NMzMLJlDw8zMkjk0zMwsmUPDzMySOTTMzCyZQ8PMzJI5NMzMLFkhH/cq6W7g1Pzl0cArEXFmhX7PA78BDgD7I6KtiHrMzKw+CgmNiPiTvmlJXwf29NP9oxHxUhF1mJlZfRUSGn0kCfhj4GNFrsfMzBqj6HMafwBsj4jnqswP4EFJKyW1F1yLmZnVaNB7GpKWA8dXmNURET/Kpy8F7uxnMedExFZJxwHLJD0bEQ9XWV870A4wbdq0wZZtZmY1UEQUs2BpJPAi8MGI2JLQ/0Zgb0QsPFzftra26Onpqb1IM7N3EUkra73gqMjDU3OBZ6sFhqSxko7qmwbOA9YWWI+ZmdWoyNCYT9mhKUmTJS3NX04CHpH0JPBL4CcR8dMC6zEzsxoVdvVURFxeoW0rMC+f3gicUdT6zcys/nxHuJmZJXNomJlZMoeGmZklc2iYmVkyh4aZmSVzaJiZWTKHhpmZJXNomJlZMoeGmZklc2iYmVkyh4aZmSVzaJiZWTKHhpmZJXNomJlZMoeGmZklc2iYmVkyh4aZmSVzaJiZWTKHhpmZJaspNCT9kaSnJb0lqa1s3pclrZe0TtInqrz/JEmPS3pO0t2SRtdSj5mZFavWPY21wB8CD5c2SjoNmA+cDpwPfFtSS4X33wx8MyJOAXYDn6uxHjMzK1BNoRERz0TEugqzLgLuioh9EfFrYD0wu7SDJAEfA36YN30fuLiWeszMrFgjC1ruFOCxktdb8rZSE4BXImJ/P33eJqkdaM9f7pO0tk61FulY4KVmF3EYw6FGcJ315jrra7jUeWqtCzhsaEhaDhxfYVZHRPyo2tsqtMUg+vx2RkQX0JXX1BMRbdX6DhXDoc7hUCO4znpznfU1nOqsdRmHDY2ImDuI5W4BTix5PRXYWtbnJeBoSSPzvY1KfczMbAgp6pLbJcB8SWMknQScAvyytENEBPAz4DN502VAtT0XMzMbAmq95PbTkrYAZwE/kfQAQEQ8Dfwb8Cvgp8AXIuJA/p6lkibni/gS8L8lrSc7x/G9xFV31VJ3Aw2HOodDjeA668111te7pk5l/+E3MzM7PN8RbmZmyRwaZmaWbMiGxnB7REm+jtX51/OSVlfp97ykNXm/mi9/G0SdN0p6saTWeVX6nZ+P73pJ1zWhzr+T9KykpyTdK+noKv2aMp6HG5/8IpC78/mPS5rRqNpKajhR0s8kPZP/Lv15hT7nStpTsj3c0Og68zr6/Tkq8/f5eD4laVaD6zu1ZIxWS3pV0jVlfZo2lpJul7Sj9P41ScdIWpb/DVwmaXyV916W93lO0mWHXVlEDMkv4HfIbkT5OdBW0n4a8CQwBjgJ2AC0VHj/vwHz8+lFwOcbWPvXgRuqzHseOLaJ43ojcO1h+rTk4zoTGJ2P92kNrvM8YGQ+fTNw81AZz5TxAf4nsCifng/c3YSf9QnArHz6KOA/K9R5LvDjRtc20J8jMA+4n+z+rg8Bjzex1hbgv4DpQ2UsgY8As4C1JW1fA67Lp6+r9DsEHANszL+Pz6fH97euIbunEcP0ESX5uv8YuLMR6yvIbGB9RGyMiDeAu8jGvWEi4sH47dMCHiO7j2eoSBmfi8i2O8i2wzn5ttEwEbEtIlbl078BnqGfpy4McRcB/xSZx8ju8TqhSbXMATZExKYmrf8QEfEw8HJZc+k2WO1v4CeAZRHxckTsBpaRPS+wqiEbGv2YArxQ8rrmR5TU2R8A2yPiuSrzA3hQ0sr80SjN8MV8F//2KrusKWPcSFeS/S+zkmaMZ8r4vN0n3w73kG2XTZEfHvsA8HiF2WdJelLS/ZJOb2hhv3W4n+NQ2ibnU/0/hUNhLPtMiohtkP0HAjiuQp8Bj2tRz55KoiHyiJJUifVeSv97GedExFZJxwHLJD2b/y+hbvqrE/gO8BWy8fgK2aG0K8sXUeG9db82O2U8JXUA+4HuKospfDwraNo2OBiSjgTuAa6JiFfLZq8iO8yyNz+/tZjsZtxGO9zPcUiMZ35u9FPAlyvMHipjORADHtemhkYMs0eUHK5eSSPJHhX/wX6WsTX/vkPSvWSHOur6Ry51XCV9F/hxhVkpY1yzhPG8DPgkMCfyA7AVllH4eFaQMj59fbbk28U4Dj18UDhJo8gCozsi/r18fmmIRMRSSd+WdGxENPThewk/x4ZskwkuAFZFxPbyGUNlLEtsl3RCRGzLD+XtqNBnC9m5mD5Tyc4jVzUcD08N5UeUzAWejYgtlWZKGivpqL5pspO9DX1ab9lx4E9XWf8TwCnKrkAbTbY7vqQR9fWRdD7ZEwM+FRG9Vfo0azxTxmcJ2XYH2Xb4H9WCryj5OZTvAc9ExDeq9Dm+71yLpNlkfxN2Na7K5J/jEuBP86uoPgTs6Tv00mBVjyQMhbEsU7oNVvsb+ABwnqTx+aHq8/K26ppxpj/xaoBPk6XgPmA78EDJvA6yq1fWAReUtC8FJufTM8nCZD3wA2BMA2q+A7i6rG0ysLSkpifzr6fJDsM0elz/GVgDPJVvVCeU15m/nkd2tc2GJtW5nuxY6+r8a1F5nc0cz0rjA9xEFnIA78m3u/X5djizCWP4YbJDDU+VjOM84Oq+7RT4Yj52T5JdcHB2E+qs+HMsq1PAt/LxXkPJFZUNrLOVLATGlbQNibEkC7JtwJv5383PkZ1Dewh4Lv9+TN63Dbit5L1X5tvpeuCKw63LjxExM7Nkw/HwlJmZNYlDw8zMkjk0zMwsmUPDzMySOTTMzCyZQ8PMzJI5NMzMLNn/B4jYFRiKhLOSAAAAAElFTkSuQmCC\n",
 79 |       "text/plain": [
 80 |        "<Figure size 432x288 with 1 Axes>"
 81 |       ]
 82 |      },
 83 |      "metadata": {
 84 |       "needs_background": "light"
 85 |      },
 86 |      "output_type": "display_data"
 87 |     },
 88 |     {
 89 |      "name": "stdout",
 90 |      "output_type": "stream",
 91 |      "text": [
 92 |       "Rotating 2.356194490192345 or 135.0 degrees\n",
 93 |       "[[-0.71 -0.71]\n",
 94 |       " [-0.71  0.71]]\n"
 95 |      ]
 96 |     },
 97 |     {
 98 |      "data": {
 99 |       "image/png": 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8GPh2Zn63pnokSZOglvs0MvN5Hdp3Asur6S3AC+v4fElSPbwjXJJUzNCQJBUzNCRJxQwNSVIxQ0OSVMzQkCQVMzQkScUMDUlSMUNDklTM0JAkFTM0JEnFDA1JUjFDQ5JUzNCQJBUzNCRJxQwNSVIxQ0OSVMzQkCQVqy00IuKjEfFQ9TfC74iI5R36XRoR90XE5oi4sq56JEndq+VvhDf5dGZe1WlhRMwEPge8GtgB/CQi1mbmz2quS5I0Af0+PLUU2JyZWzLzKeArwIo+1yRJ6qDu0PhARNwVEddGxNw2yxcADzbN76jaDhMRgxExGhGju3fvrqNWSdJRdBUaEXFzRNzd5msF8HngHOBFwMPAp9qtok1btvuszBzOzIHMHJg/f343ZUuSJqircxqZeXFJv4j4W+BbbRbtAM5umj8L2NlNTZKk+tR59dQZTbNvAO5u0+0nwLkR8dyIOAFYCaytqyZJUnfqvHrqzyLiRTQON20F/hAgIs4ErsnM5Zl5ICI+ANwIzASuzcyf1liTJKkLtYVGZr6tQ/tOYHnT/DpgXV11SJImT78vuZUkTSOGhiSpmKEhSSpmaEiSihkakqRihoYkqZihIUkqZmhIkooZGpKkYoaGJKmYoSFJKmZoSJKKGRqSpGKGhiSpmKEhSSpmaEiSihkakqRihoYkqVgtf+41Ir4KnFfNPhv4RWa+qE2/rcAvgYPAgcwcqKMeSdLkqCU0MvM/j09HxKeAvUfo/srM/HkddUiSJlctoTEuIgJ4C/CqOj9HktQbdZ/T+H1gV2be32F5AjdFxIaIGKy5FklSlya8pxERNwOnt1k0lJnfqKbfCnz5CKt5eWbujIhTgfURcW9m3trh8waBQYCFCxdOtGxJUhciM+tZccRxwEPASzJzR0H/jwL7MvOqo/UdGBjI0dHR7ouUpN8iEbGh2wuO6jw8dTFwb6fAiIg5EXHS+DRwCXB3jfVIkrpUZ2ispOXQVEScGRHrqtnTgB9ExJ3Aj4FvZ+Z3a6xHktSl2q6eysx3tGnbCSyvprcAL6zr8yVJk887wiVJxQwNSVIxQ0OSVMzQkCQVMzQkScUMDUlSMUNDklTM0JAkFTM0JEnFDA1JUjFDQ5JUzNCQJBUzNCRJxQwNSVIxQ0OSVMzQkCQVMzQkScUMDUlSMUNDklSsq9CIiDdHxE8j4tcRMdCy7EMRsTki7ouI13R4/3Mj4vaIuD8ivhoRJ3RTjySpXt3uadwN/Cfg1ubGiDgfWAm8ALgU+JuImNnm/Z8EPp2Z5wKPAe/ush5JUo26Co3MvCcz72uzaAXwlczcn5n/F9gMLG3uEBEBvAr4h6rpi8Bl3dQjSarXcTWtdwHwo6b5HVVbs3nALzLzwBH6PC0iBoHBanZ/RNw9SbXW6TnAz/tdxFFMhxrBOiebdU6u6VLned2u4KihERE3A6e3WTSUmd/o9LY2bTmBPr9ZkDkMDFc1jWbmQKe+U8V0qHM61AjWOdmsc3JNpzq7XcdRQyMzL57AencAZzfNnwXsbOnzc+DZEXFctbfRro8kaQqp65LbtcDKiJgVEc8FzgV+3NwhMxP4HvCmqulyoNOeiyRpCuj2kts3RMQO4ELg2xFxI0Bm/hT4GvAz4LvA+zPzYPWedRFxZrWKPwb+W0RspnGO4wuFHz3cTd09NB3qnA41gnVONuucXL81dUbjF35Jko7OO8IlScUMDUlSsSkbGtPtESXVZ9xRfW2NiDs69NsaEZuqfl1f/jaBOj8aEQ811bq8Q79Lq/HdHBFX9qHOP4+IeyPiroi4ISKe3aFfX8bzaONTXQTy1Wr57RGxuFe1NdVwdkR8LyLuqf4v/VGbPq+IiL1N28NHel1nVccRv4/R8FfVeN4VERf0uL7zmsbojoh4PCI+2NKnb2MZEddGxCPN969FxCkRsb76Gbg+IuZ2eO/lVZ/7I+Lyo35YZk7JL+Df0rgR5fvAQFP7+cCdwCzgucADwMw27/8asLKavhp4Xw9r/xTwkQ7LtgLP6eO4fhS44ih9ZlbjugQ4oRrv83tc5yXAcdX0J4FPTpXxLBkf4L8CV1fTK4Gv9uF7fQZwQTV9EvAvbep8BfCtXtd2rN9HYDnwHRr3d70UuL2Ptc4E/hVYNFXGEvgD4ALg7qa2PwOurKavbPd/CDgF2FK9zq2m5x7ps6bsnkZO00eUVJ/9FuDLvfi8miwFNmfmlsx8CvgKjXHvmcy8KX/ztIAf0biPZ6ooGZ8VNLY7aGyHy6pto2cy8+HM3FhN/xK4hyM8dWGKWwF8KRt+ROMerzP6VMsy4IHM3Nanzz9MZt4KPNrS3LwNdvoZ+BpgfWY+mpmPAetpPC+woykbGkewAHiwab7rR5RMst8HdmXm/R2WJ3BTRGyoHo3SDx+odvGv7bDLWjLGvfQuGr9lttOP8SwZn6f7VNvhXhrbZV9Uh8deDNzeZvGFEXFnRHwnIl7Q08J+42jfx6m0Ta6k8y+FU2Esx52WmQ9D4xcI4NQ2fY55XOt69lSRmCKPKClVWO9bOfJexsszc2dEnAqsj4h7q98SJs2R6gQ+D3ycxnh8nMahtHe1rqLNeyf92uyS8YyIIeAAMNJhNbWPZxt92wYnIiJOBK4HPpiZj7cs3kjjMMu+6vzWGho34/ba0b6PU2I8q3Ojrwc+1GbxVBnLY3HM49rX0Mhp9oiSo9UbEcfReFT8S46wjp3V6yMRcQONQx2T+kOudFwj4m+Bb7VZVDLGXSsYz8uB1wHLsjoA22YdtY9nGyXjM95nR7VdnMzhhw9qFxHH0wiMkcz8x9blzSGSmesi4m8i4jmZ2dOH7xV8H3uyTRZ4LbAxM3e1LpgqY9lkV0SckZkPV4fyHmnTZweNczHjzqJxHrmj6Xh4aio/ouRi4N7M3NFuYUTMiYiTxqdpnOzt6dN6W44Dv6HD5/8EODcaV6CdQGN3fG0v6hsXEZfSeGLA6zNzrEOffo1nyfispbHdQWM7/KdOwVeX6hzKF4B7MvMvOvQ5ffxcS0QspfEzYU/vqiz+Pq4F3l5dRfVSYO/4oZce63gkYSqMZYvmbbDTz8AbgUsiYm51qPqSqq2zfpzpL7wa4A00UnA/sAu4sWnZEI2rV+4DXtvUvg44s5peQiNMNgNfB2b1oObrgPe2tJ0JrGuq6c7q66c0DsP0elz/DtgE3FVtVGe01lnNL6dxtc0DfapzM41jrXdUX1e31tnP8Ww3PsDHaIQcwO9U293majtc0ocxvIjGoYa7msZxOfDe8e0U+EA1dnfSuODgZX2os+33saXOAD5Xjfcmmq6o7GGds2mEwMlNbVNiLGkE2cPAr6qfm++mcQ7tFuD+6vWUqu8AcE3Te99VbaebgXce7bN8jIgkqdh0PDwlSeoTQ0OSVMzQkCQVMzQkScUMDUlSMUNDklTM0JAkFfv/OranSOyMnxsAAAAASUVORK5CYII=\n",
100 |       "text/plain": [
101 |        "<Figure size 432x288 with 1 Axes>"
102 |       ]
103 |      },
104 |      "metadata": {
105 |       "needs_background": "light"
106 |      },
107 |      "output_type": "display_data"
108 |     },
109 |     {
110 |      "name": "stdout",
111 |      "output_type": "stream",
112 |      "text": [
113 |       "[[6.71 0.  ]\n",
114 |       " [0.   2.24]]\n",
115 |       "Stretching x-axis by 6.70820393249937\n",
116 |       "Stretching y-axis by 2.2360679774997894\n"
117 |      ]
118 |     },
119 |     {
120 |      "data": {
121 |       "image/png": 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\n",
122 |       "text/plain": [
123 |        "<Figure size 432x288 with 1 Axes>"
124 |       ]
125 |      },
126 |      "metadata": {
127 |       "needs_background": "light"
128 |      },
129 |      "output_type": "display_data"
130 |     },
131 |     {
132 |      "name": "stdout",
133 |      "output_type": "stream",
134 |      "text": [
135 |       "Rotating -1.8925468811915387 or -108.43494882292201 degrees\n",
136 |       "[[-0.32 -0.95]\n",
137 |       " [-0.95  0.32]]\n"
138 |      ]
139 |     },
140 |     {
141 |      "data": {
142 |       "image/png": 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\n",
143 |       "text/plain": [
144 |        "<Figure size 432x288 with 1 Axes>"
145 |       ]
146 |      },
147 |      "metadata": {
148 |       "needs_background": "light"
149 |      },
150 |      "output_type": "display_data"
151 |     },
152 |     {
153 |      "name": "stdout",
154 |      "output_type": "stream",
155 |      "text": [
156 |       "Original transformation\n",
157 |       "[[3 0]\n",
158 |       " [4 5]]\n"
159 |      ]
160 |     },
161 |     {
162 |      "data": {
163 |       "image/png": 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\n",
164 |       "text/plain": [
165 |        "<Figure size 432x288 with 1 Axes>"
166 |       ]
167 |      },
168 |      "metadata": {
169 |       "needs_background": "light"
170 |      },
171 |      "output_type": "display_data"
172 |     }
173 |    ],
174 |    "source": [
175 |     "#A = np.array([[0,1],[-2,-3]])\n",
176 |     "#A = np.array([[2,2],[1,-1]])\n",
177 |     "#A = np.array([[4,8],\n",
178 |     "#              [-5,3]])\n",
179 |     "A = np.array([[3,0],\n",
180 |     "              [4,5]])\n",
181 |     "\n",
182 |     "U,S,V = plot_circles(A, xlim=10, ylim=10, n=8)"
183 |    ]
184 |   },
185 |   {
186 |    "cell_type": "code",
187 |    "execution_count": 8,
188 |    "metadata": {},
189 |    "outputs": [
190 |     {
191 |      "name": "stdout",
192 |      "output_type": "stream",
193 |      "text": [
194 |       "[[ 0.31622777  0.9486833 ]\n",
195 |       " [ 0.9486833  -0.31622777]]\n",
196 |       "[[6.70820393 0.        ]\n",
197 |       " [0.         2.23606798]]\n",
198 |       "[[ 0.70710678  0.70710678]\n",
199 |       " [ 0.70710678 -0.70710678]]\n"
200 |      ]
201 |     }
202 |    ],
203 |    "source": [
204 |     "print(-U)\n",
205 |     "sigma = np.diag(S)\n",
206 |     "print(sigma)\n",
207 |     "print(-V)"
208 |    ]
209 |   },
210 |   {
211 |    "cell_type": "code",
212 |    "execution_count": 9,
213 |    "metadata": {},
214 |    "outputs": [
215 |     {
216 |      "data": {
217 |       "text/plain": [
218 |        "array([0, 5])"
219 |       ]
220 |      },
221 |      "execution_count": 9,
222 |      "metadata": {},
223 |      "output_type": "execute_result"
224 |     }
225 |    ],
226 |    "source": [
227 |     "np.dot(A, np.array([0,1]))"
228 |    ]
229 |   },
230 |   {
231 |    "cell_type": "code",
232 |    "execution_count": 10,
233 |    "metadata": {},
234 |    "outputs": [
235 |     {
236 |      "data": {
237 |       "text/plain": [
238 |        "array([3, 0])"
239 |       ]
240 |      },
241 |      "execution_count": 10,
242 |      "metadata": {},
243 |      "output_type": "execute_result"
244 |     }
245 |    ],
246 |    "source": [
247 |     "np.dot(np.array([1,0]), A)"
248 |    ]
249 |   },
250 |   {
251 |    "cell_type": "code",
252 |    "execution_count": 11,
253 |    "metadata": {},
254 |    "outputs": [
255 |     {
256 |      "data": {
257 |       "text/plain": [
258 |        "array([4, 5])"
259 |       ]
260 |      },
261 |      "execution_count": 11,
262 |      "metadata": {},
263 |      "output_type": "execute_result"
264 |     }
265 |    ],
266 |    "source": [
267 |     "np.dot(np.array([0,1]), A)"
268 |    ]
269 |   },
270 |   {
271 |    "cell_type": "code",
272 |    "execution_count": 12,
273 |    "metadata": {},
274 |    "outputs": [
275 |     {
276 |      "data": {
277 |       "text/plain": [
278 |        "array([[ 3.        ,  0.        , -3.        ,  0.        ,  2.12132034,\n",
279 |        "         2.12132034, -2.12132034, -2.12132034],\n",
280 |        "       [ 4.        ,  5.        , -4.        , -5.        ,  6.36396103,\n",
281 |        "        -0.70710678,  0.70710678, -6.36396103]])"
282 |       ]
283 |      },
284 |      "execution_count": 12,
285 |      "metadata": {},
286 |      "output_type": "execute_result"
287 |     }
288 |    ],
289 |    "source": [
290 |     "#x = np.array([[1,0,-1,0],[0,1,0,-1]])\n",
291 |     "x = np.array([[1,0],\n",
292 |     "              [0,1],\n",
293 |     "              [-1,0],\n",
294 |     "              [0,-1],\n",
295 |     "              [1/np.sqrt(2), 1/np.sqrt(2)],\n",
296 |     "              [1/np.sqrt(2), -1/np.sqrt(2)],\n",
297 |     "              [-1/np.sqrt(2), 1/np.sqrt(2)],\n",
298 |     "              [-1/np.sqrt(2), -1/np.sqrt(2)]])\n",
299 |     "\n",
300 |     "np.dot(A,x.T)"
301 |    ]
302 |   },
303 |   {
304 |    "cell_type": "code",
305 |    "execution_count": 13,
306 |    "metadata": {},
307 |    "outputs": [
308 |     {
309 |      "data": {
310 |       "text/plain": [
311 |        "array([[-0.70710678, -0.70710678,  0.70710678,  0.70710678, -1.        ,\n",
312 |        "         0.        ,  0.        ,  1.        ],\n",
313 |        "       [-0.70710678,  0.70710678,  0.70710678, -0.70710678,  0.        ,\n",
314 |        "        -1.        ,  1.        ,  0.        ]])"
315 |       ]
316 |      },
317 |      "execution_count": 13,
318 |      "metadata": {},
319 |      "output_type": "execute_result"
320 |     }
321 |    ],
322 |    "source": [
323 |     "vx = np.dot(V,x.T)\n",
324 |     "vx"
325 |    ]
326 |   },
327 |   {
328 |    "cell_type": "code",
329 |    "execution_count": 14,
330 |    "metadata": {},
331 |    "outputs": [
332 |     {
333 |      "data": {
334 |       "text/plain": [
335 |        "array([[-4.74341649, -4.74341649,  4.74341649,  4.74341649, -6.70820393,\n",
336 |        "         0.        ,  0.        ,  6.70820393],\n",
337 |        "       [-1.58113883,  1.58113883,  1.58113883, -1.58113883,  0.        ,\n",
338 |        "        -2.23606798,  2.23606798,  0.        ]])"
339 |       ]
340 |      },
341 |      "execution_count": 14,
342 |      "metadata": {},
343 |      "output_type": "execute_result"
344 |     }
345 |    ],
346 |    "source": [
347 |     "svx = np.dot(sigma,vx)\n",
348 |     "svx"
349 |    ]
350 |   },
351 |   {
352 |    "cell_type": "code",
353 |    "execution_count": 15,
354 |    "metadata": {},
355 |    "outputs": [
356 |     {
357 |      "data": {
358 |       "text/plain": [
359 |        "array([[ 3.00000000e+00,  8.88178420e-16, -3.00000000e+00,\n",
360 |        "        -8.88178420e-16,  2.12132034e+00,  2.12132034e+00,\n",
361 |        "        -2.12132034e+00, -2.12132034e+00],\n",
362 |        "       [ 4.00000000e+00,  5.00000000e+00, -4.00000000e+00,\n",
363 |        "        -5.00000000e+00,  6.36396103e+00, -7.07106781e-01,\n",
364 |        "         7.07106781e-01, -6.36396103e+00]])"
365 |       ]
366 |      },
367 |      "execution_count": 15,
368 |      "metadata": {},
369 |      "output_type": "execute_result"
370 |     }
371 |    ],
372 |    "source": [
373 |     "usvx = np.dot(U,svx)\n",
374 |     "usvx"
375 |    ]
376 |   },
377 |   {
378 |    "cell_type": "code",
379 |    "execution_count": 16,
380 |    "metadata": {},
381 |    "outputs": [
382 |     {
383 |      "data": {
384 |       "text/plain": [
385 |        "71.56505117707799"
386 |       ]
387 |      },
388 |      "execution_count": 16,
389 |      "metadata": {},
390 |      "output_type": "execute_result"
391 |     }
392 |    ],
393 |    "source": [
394 |     "np.arctan(3)*180/np.pi"
395 |    ]
396 |   },
397 |   {
398 |    "cell_type": "code",
399 |    "execution_count": 17,
400 |    "metadata": {},
401 |    "outputs": [
402 |     {
403 |      "data": {
404 |       "text/plain": [
405 |        "163.23296235748467"
406 |       ]
407 |      },
408 |      "execution_count": 17,
409 |      "metadata": {},
410 |      "output_type": "execute_result"
411 |     }
412 |    ],
413 |    "source": [
414 |     "73*np.sqrt(5)"
415 |    ]
416 |   },
417 |   {
418 |    "cell_type": "code",
419 |    "execution_count": 18,
420 |    "metadata": {},
421 |    "outputs": [
422 |     {
423 |      "data": {
424 |       "text/plain": [
425 |        "0.9486832980505138"
426 |       ]
427 |      },
428 |      "execution_count": 18,
429 |      "metadata": {},
430 |      "output_type": "execute_result"
431 |     }
432 |    ],
433 |    "source": [
434 |     "3/np.sqrt(10)"
435 |    ]
436 |   },
437 |   {
438 |    "cell_type": "code",
439 |    "execution_count": 19,
440 |    "metadata": {},
441 |    "outputs": [
442 |     {
443 |      "data": {
444 |       "text/plain": [
445 |        "array([[1.8, 1.2],\n",
446 |        "       [4.4, 4.6]])"
447 |       ]
448 |      },
449 |      "execution_count": 19,
450 |      "metadata": {},
451 |      "output_type": "execute_result"
452 |     }
453 |    ],
454 |    "source": [
455 |     "sigma_2 = np.diag([3*np.sqrt(5), 1/np.sqrt(5)])\n",
456 |     "E = np.dot(np.dot(U,sigma_2),V.T)\n",
457 |     "E"
458 |    ]
459 |   },
460 |   {
461 |    "cell_type": "code",
462 |    "execution_count": 20,
463 |    "metadata": {},
464 |    "outputs": [
465 |     {
466 |      "data": {
467 |       "image/png": 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\n",
468 |       "text/plain": [
469 |        "<Figure size 432x288 with 1 Axes>"
470 |       ]
471 |      },
472 |      "metadata": {
473 |       "needs_background": "light"
474 |      },
475 |      "output_type": "display_data"
476 |     },
477 |     {
478 |      "name": "stdout",
479 |      "output_type": "stream",
480 |      "text": [
481 |       "Rotating 2.356194490192345 or 135.0 degrees\n",
482 |       "[[-0.71 -0.71]\n",
483 |       " [-0.71  0.71]]\n"
484 |      ]
485 |     },
486 |     {
487 |      "data": {
488 |       "image/png": 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\n",
489 |       "text/plain": [
490 |        "<Figure size 432x288 with 1 Axes>"
491 |       ]
492 |      },
493 |      "metadata": {
494 |       "needs_background": "light"
495 |      },
496 |      "output_type": "display_data"
497 |     },
498 |     {
499 |      "name": "stdout",
500 |      "output_type": "stream",
501 |      "text": [
502 |       "[[6.71 0.  ]\n",
503 |       " [0.   0.45]]\n",
504 |       "Stretching x-axis by 6.708203932499367\n",
505 |       "Stretching y-axis by 0.447213595499958\n"
506 |      ]
507 |     },
508 |     {
509 |      "data": {
510 |       "image/png": 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\n",
511 |       "text/plain": [
512 |        "<Figure size 432x288 with 1 Axes>"
513 |       ]
514 |      },
515 |      "metadata": {
516 |       "needs_background": "light"
517 |      },
518 |      "output_type": "display_data"
519 |     },
520 |     {
521 |      "name": "stdout",
522 |      "output_type": "stream",
523 |      "text": [
524 |       "Rotating -1.8925468811915391 or -108.43494882292204 degrees\n",
525 |       "[[-0.32 -0.95]\n",
526 |       " [-0.95  0.32]]\n"
527 |      ]
528 |     },
529 |     {
530 |      "data": {
531 |       "image/png": 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\n",
532 |       "text/plain": [
533 |        "<Figure size 432x288 with 1 Axes>"
534 |       ]
535 |      },
536 |      "metadata": {
537 |       "needs_background": "light"
538 |      },
539 |      "output_type": "display_data"
540 |     },
541 |     {
542 |      "name": "stdout",
543 |      "output_type": "stream",
544 |      "text": [
545 |       "Original transformation\n",
546 |       "[[1.8 1.2]\n",
547 |       " [4.4 4.6]]\n"
548 |      ]
549 |     },
550 |     {
551 |      "data": {
552 |       "text/plain": [
553 |        "(array([[-0.31622777, -0.9486833 ],\n",
554 |        "        [-0.9486833 ,  0.31622777]]),\n",
555 |        " array([6.70820393, 0.4472136 ]),\n",
556 |        " array([[-0.70710678, -0.70710678],\n",
557 |        "        [-0.70710678,  0.70710678]]))"
558 |       ]
559 |      },
560 |      "execution_count": 20,
561 |      "metadata": {},
562 |      "output_type": "execute_result"
563 |     },
564 |     {
565 |      "data": {
566 |       "image/png": 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\n",
567 |       "text/plain": [
568 |        "<Figure size 432x288 with 1 Axes>"
569 |       ]
570 |      },
571 |      "metadata": {
572 |       "needs_background": "light"
573 |      },
574 |      "output_type": "display_data"
575 |     }
576 |    ],
577 |    "source": [
578 |     "plot_circles(E, 10, 10)"
579 |    ]
580 |   },
581 |   {
582 |    "cell_type": "code",
583 |    "execution_count": 21,
584 |    "metadata": {},
585 |    "outputs": [
586 |     {
587 |      "data": {
588 |       "text/plain": [
589 |        "63.50433056099403"
590 |       ]
591 |      },
592 |      "execution_count": 21,
593 |      "metadata": {},
594 |      "output_type": "execute_result"
595 |     }
596 |    ],
597 |    "source": [
598 |     "142/np.sqrt(5)"
599 |    ]
600 |   },
601 |   {
602 |    "cell_type": "code",
603 |    "execution_count": 22,
604 |    "metadata": {},
605 |    "outputs": [
606 |     {
607 |      "data": {
608 |       "text/plain": [
609 |        "array([[1.5, 1.5],\n",
610 |        "       [4.5, 4.5]])"
611 |       ]
612 |      },
613 |      "execution_count": 22,
614 |      "metadata": {},
615 |      "output_type": "execute_result"
616 |     }
617 |    ],
618 |    "source": [
619 |     "sigma_3 = np.diag([3*np.sqrt(5), 0])\n",
620 |     "F = np.dot(np.dot(U,sigma_3),V.T)\n",
621 |     "F"
622 |    ]
623 |   },
624 |   {
625 |    "cell_type": "code",
626 |    "execution_count": null,
627 |    "metadata": {},
628 |    "outputs": [],
629 |    "source": []
630 |   }
631 |  ],
632 |  "metadata": {
633 |   "kernelspec": {
634 |    "display_name": "Python 3",
635 |    "language": "python",
636 |    "name": "python3"
637 |   },
638 |   "language_info": {
639 |    "codemirror_mode": {
640 |     "name": "ipython",
641 |     "version": 3
642 |    },
643 |    "file_extension": ".py",
644 |    "mimetype": "text/x-python",
645 |    "name": "python",
646 |    "nbconvert_exporter": "python",
647 |    "pygments_lexer": "ipython3",
648 |    "version": "3.7.6"
649 |   }
650 |  },
651 |  "nbformat": 4,
652 |  "nbformat_minor": 4
653 | }
654 | 


--------------------------------------------------------------------------------