{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The SVD decomposition of a symmetric matrix ##"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let S be a symmetric matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "S=np.array([[2, -1, 3], [-1, 2, -1], [3, -1, 2]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "D, P=np.linalg.eigh(S)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1.        ,  1.43844719,  5.56155281])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 7.07106781e-01,  2.60956474e-01, -6.57192300e-01],\n",
       "       [ 2.08159786e-16,  9.29410263e-01,  3.69048184e-01],\n",
       "       [-7.07106781e-01,  2.60956474e-01, -6.57192300e-01]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$PDP^T$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2., -1.,  3.],\n",
       "       [-1.,  2., -1.],\n",
       "       [ 3., -1.,  2.]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot(P, np.dot(np.diag(D), P.T))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Singular decomposition of the same matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "U, Sigma, Vt =np.linalg.svd(S)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([5.56155281, 1.43844719, 1.        ])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sigma"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Comparing $\\Sigma$ to D we notice that $\\Sigma=\\pi (|D|)$, where $\\pi$ is the permutation $\\pi(0)=2, \\pi(1)=1, \\pi(2)=0$,\n",
    "i.e. $\\pi(|\\lambda_0|)=\\sigma_2$, etc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-6.57192300e-01,  2.60956474e-01,  7.07106781e-01],\n",
       "       [ 3.69048184e-01,  9.29410263e-01, -1.59154867e-16],\n",
       "       [-6.57192300e-01,  2.60956474e-01, -7.07106781e-01]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The columns in U are the column in P permuted by $\\pi$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-6.57192300e-01,  3.69048184e-01, -6.57192300e-01],\n",
       "       [ 2.60956474e-01,  9.29410263e-01,  2.60956474e-01],\n",
       "       [-7.07106781e-01, -3.96348516e-16,  7.07106781e-01]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Vt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Vt=U^T`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-6.57192300e-01,  3.69048184e-01, -6.57192300e-01],\n",
       "       [ 2.60956474e-01,  9.29410263e-01,  2.60956474e-01],\n",
       "       [ 7.07106781e-01, -1.59154867e-16, -7.07106781e-01]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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