qc-mentorship-program/Task2/mathematics.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Eigenvalues and Eigenvectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.linalg as la"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": []
   },
   "source": [
    "Let A be a sqaure matrix. A non-zero vector v is an eigenvector for A with eigenvalue  if\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": []
   },
   "source": [
    "Rearranging the equation, we see v is a solution of homogenous system of equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": []
   },
   "source": [
    "where I is the identity matrix of size n. Eigenvalues of A are roots of the characteristic polynomial"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "tags": []
   },
   "source": [
    "The function `scipy.linalg.eig` computes eigenvalues and eigenvectors of a square matrix "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1  0]\n",
      " [ 0 -1]]\n"
     ]
    }
   ],
   "source": [
    "# Let's consider a simple example\n",
    "A = np.array([[1,0],[0,-1]])\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "# The function la.eig returns eigenvalues and eigenvectors as tuples\n",
    "eigvals, eigvecs = la.eig(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.+0.j -1.+0.j]\n"
     ]
    }
   ],
   "source": [
    "# The eigenvalues of A are\n",
    "print(eigvals)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1. 0.]\n",
      " [0. 1.]]\n"
     ]
    }
   ],
   "source": [
    "# The eigenvectors of A are\n",
    "print(eigvecs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1. -1.]\n"
     ]
    }
   ],
   "source": [
    "# If we know the eigenvalues are real numbers, A is symmetrics, then we use the method `.real` to convert into real numbers\n",
    "print(eigvals.real)"
   ]
  },
  {
   "attachments": {
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    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Symmetric Matrices\n",
    "Symmetric matrix is a square matrix that is qual to its taranspose\n",
    "\n",
    "![image.png](attachment:image.png)\n",
    "\n",
    "- The entries of a symmetric matrix are symmetrics with respect to the main diagonal\n",
    "- eigenvalues are real\n",
    "- eigenvectors are always orthogonal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1  7  3]\n",
      " [ 7  4 -5]\n",
      " [ 3 -5  6]]\n"
     ]
    }
   ],
   "source": [
    "A = np.array([[1, 7, 3],\n",
    "              [7, 4, -5],\n",
    "              [3, -5, 6]])\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1  7  3]\n",
      " [ 7  4 -5]\n",
      " [ 3 -5  6]]\n"
     ]
    }
   ],
   "source": [
    "# Transpose of A\n",
    "print(A.T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "eigvals, eigvecs = la.eig(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-7.00792852+0.j  7.03594585+0.j 10.97198268+0.j]\n"
     ]
    }
   ],
   "source": [
    "# Eigenvalues have zero imaginary part\n",
    "print(eigvals)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.68415383 -0.62878419 -0.36954564]\n",
      " [-0.61391034 -0.22292725 -0.75724338]\n",
      " [-0.39376087 -0.74493885  0.53853365]]\n"
     ]
    }
   ],
   "source": [
    "print(eigvecs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0\n"
     ]
    }
   ],
   "source": [
    "# Check that eigenvectors are orthogonal\n",
    "# The dot product of eigenvectors  and  is zero \n",
    "v1 = eigvecs[:,0] # First column is the first eigenvector\n",
    "v2 = eigvecs[:,1] # Second column is the second eigenvector\n",
    "print(v1 @ v2)\n"
   ]
  }
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