{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "m = 2\n",
    "n = 2\n",
    "\n",
    "A = np.array([[3, 7],\n",
    "              [5, 2]])\n",
    "\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.linalg import svd\n",
    "\n",
    "U, d, Vt = svd(A)\n",
    "U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Vt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Create the diagonal matrix D from vector d\n",
    "D = np.zeros((m, n))\n",
    "\n",
    "for i in range(min(m, n)):\n",
    "    D[i, i] = d[i]\n",
    "\n",
    "D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "U@D@Vt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy.linalg import norm\n",
    "\n",
    "def is_orthonormal(name, A):\n",
    "    print()\n",
    "    print(f'Show that matrix {name} is orthonormal:')\n",
    "    print()\n",
    "\n",
    "    col = []\n",
    "\n",
    "    for i in range(n):\n",
    "        col += [A.T[i]]\n",
    "        print(f'Norm of column {i} = {norm(col[i])} ')\n",
    "\n",
    "    print()\n",
    "\n",
    "    for i in range(n):\n",
    "        for j in range(i + 1, n):\n",
    "            print(f'column {i} @ column {j} = {col[i]@col[j]}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "is_orthonormal('U', U)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "is_orthonormal('V', Vt.T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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