{
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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "def plot_vectors(vecs, xlim, ylim):\n",
    "    \"\"\"\n",
    "    Plot set of vectors.\n",
    "    @parm vecs   [array-like] Coordinates of the vectors to plot. \n",
    "                 Each vector is in an array. For instance: \n",
    "                 [[1, 3], [2, 2]] can be used to plot 2 vectors.\n",
    "    @return      matplotlib.figure.Figure of the vectors\n",
    "    \"\"\"\n",
    "    plt.figure()\n",
    "    plt.axvline(x=0, color='#A9A9A9', zorder=0)\n",
    "    plt.axhline(y=0, color='#A9A9A9', zorder=0)\n",
    "    plt.xlim(xlim[0], xlim[1])\n",
    "    plt.ylim(ylim[0], ylim[1])\n",
    "    \n",
    "    vector_colors = ['#1190FF', '#FF9A13', '#000000']  # blue, orange, black\n",
    "\n",
    "    for i in range(len(vecs)):\n",
    "        x = np.concatenate([[0,0], vecs[i].flatten()])\n",
    "        plt.quiver([x[0]], [x[1]], [x[2]], [x[3]],\n",
    "                   angles='xy', scale_units='xy', scale=1, \n",
    "                   color=vector_colors[i])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.array([[2], \n",
    "              [1]])\n",
    "\n",
    "print('x =')\n",
    "print(x)\n",
    "\n",
    "plot_vectors([x], \n",
    "             xlim=(-1, 4), ylim=(-1, 4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = np.array([[-1,  3], \n",
    "              [ 2, -2]])\n",
    "x = np.array([[2], \n",
    "              [1]])\n",
    "\n",
    "Ax = A@x\n",
    "\n",
    "print('A =')\n",
    "print(A)\n",
    "print('x =')\n",
    "print(x)\n",
    "print('Ax =')\n",
    "print(Ax)\n",
    "\n",
    "plot_vectors([x, Ax], \n",
    "             xlim=(-1, 4), ylim=(-1, 4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.array([[1], \n",
    "              [1]])\n",
    "\n",
    "print('x =')\n",
    "print(x)\n",
    "\n",
    "plot_vectors([x], \n",
    "             xlim=(-1, 7), \n",
    "             ylim=(-1, 7))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = np.array([[5, 1], \n",
    "              [3, 3]])\n",
    "x = np.array([[1], \n",
    "              [1]])\n",
    "\n",
    "λ = 6\n",
    "Ax = A@x\n",
    "λx = λ*x\n",
    "\n",
    "print('A =')\n",
    "print(A)\n",
    "print('x =')\n",
    "print(x)\n",
    "print('Ax =')\n",
    "print(A@x)\n",
    "print(f'λ = {λ}')\n",
    "print('λx =')\n",
    "print(λx)\n",
    "\n",
    "plot_vectors([Ax, x], \n",
    "             xlim=(-1, 7), \n",
    "             ylim=(-1, 7))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.array([[ 1], \n",
    "              [-3]])\n",
    "\n",
    "print('x =')\n",
    "print(x)\n",
    "\n",
    "plot_vectors([x], \n",
    "             xlim=(-1, 3), \n",
    "             ylim=(-7, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = np.array([[5, 1], \n",
    "              [3, 3]])\n",
    "x = np.array([[ 1], \n",
    "              [-3]])\n",
    "\n",
    "λ = 2\n",
    "Ax = A@x\n",
    "λx = λ*x\n",
    "\n",
    "print('A =')\n",
    "print(A)\n",
    "print('x =')\n",
    "print(x)\n",
    "print('Ax =')\n",
    "print(A@x)\n",
    "print(f'λ = {λ}')\n",
    "print('λx =')\n",
    "print(λx)\n",
    "\n",
    "plot_vectors([Ax, x], \n",
    "             xlim=(-1, 3), \n",
    "             ylim=(-7, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%precision 2\n",
    "from numpy.linalg import norm\n",
    "\n",
    "A = np.array([[5, 1], \n",
    "              [3, 3]])\n",
    "x = np.array([[ 1], \n",
    "              [-3]])\n",
    "\n",
    "x_normed = x/norm(x)\n",
    "\n",
    "λ = 2\n",
    "Ax_normed = A@x_normed\n",
    "λx_normed = λ*x_normed\n",
    "\n",
    "print('A =')\n",
    "print(A)\n",
    "print('x_normed =')\n",
    "print(x_normed)\n",
    "print('Ax_normed =')\n",
    "print(A@x_normed)\n",
    "print(f'λ = {λ}')\n",
    "print('λx_normed =')\n",
    "print(λx_normed)\n",
    "\n",
    "plot_vectors([Ax, x, x_normed], \n",
    "             xlim=(-1, 3), \n",
    "             ylim=(-7, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%precision 2\n",
    "\n",
    "import numpy.linalg\n",
    "\n",
    "A = np.array([[5, 1], \n",
    "              [3, 3]])\n",
    "\n",
    "λ, v = np.linalg.eig(A)\n",
    "    \n",
    "print(f'eigenvalues: {λ}')\n",
    "print('eigenvectors:')\n",
    "print(v)\n",
    "print()\n",
    "print(f'Av = {A@v[:, 1]}')\n",
    "print(f'λv = {λ[1]*v[:, 1]}')\n",
    "\n",
    "plot_vectors([Ax, x, v[:, 1]], \n",
    "             xlim=(-1, 3), \n",
    "             ylim=(-7, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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