{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Le module numpy n'est pas présent dans toutes les distributions de Python."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2]\n",
      " [3 5]]  = A\n"
     ]
    }
   ],
   "source": [
    "A = np.array([[1, 2],[3, 5]])\n",
    "print(A,\" = A\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Matrice unité ou identité :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.  0.]\n",
      " [ 0.  1.]]\n"
     ]
    }
   ],
   "source": [
    "I=np.eye(2)\n",
    "print(I)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  5.   4.]\n",
      " [  6.  13.]]  = 2A+3*I\n"
     ]
    }
   ],
   "source": [
    "print( 2*A + 3*I, \" = 2A+3*I\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Remarque :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 5  7]\n",
      " [ 9 13]]\n"
     ]
    }
   ],
   "source": [
    " print(2*A + 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.  0.  0.]\n",
      " [ 0.  0.  0.]] [[1 0 0]\n",
      " [0 2 0]\n",
      " [0 0 3]] [[ 1.  1.  1.  1.]] [ 0.75964575  0.25981134  0.61308799  0.25851246  0.96780174]\n"
     ]
    }
   ],
   "source": [
    "print(np.zeros((2,3)) , np.diag(np.array([1,2,3])),np.ones((1,4)), np.random.rand(5) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Addition de matrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2]\n",
      " [3 5]]  +  [[ 0  5]\n",
      " [ 2 -1]]  =  [[1 7]\n",
      " [5 4]]\n"
     ]
    }
   ],
   "source": [
    "B=np.array([[0,5],[2,-1]])\n",
    "print(A,\" + \", B , \" = \" , A+B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Produit de matrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 4  3]\n",
      " [10 10]]\n"
     ]
    }
   ],
   "source": [
    "print(np.dot(A,B))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Comparer avec :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0 10]\n",
      " [ 6 -5]]\n"
     ]
    }
   ],
   "source": [
    "print(A*B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Produit scalaire de vecteurs :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "30"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V1=np.array([3,2])\n",
    "V2=np.array([4,9])\n",
    "np.vdot(V1,V2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Transposée"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 3],\n",
       "       [2, 5]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.transpose()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Inverse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-5.,  2.],\n",
       "       [ 3., -1.]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from numpy import linalg\n",
    "linalg.inv(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Puissance de matrice"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 43,  74],\n",
       "       [111, 191]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "linalg.matrix_power(A,3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Valeurs propres et vecteurs propres"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  5.82842712e+00   1.71572875e-01   3.22904205e-14  -2.59045551e-16]\n",
      "[[-0.54003331 -0.23856181 -0.21018944 -0.2015738 ]\n",
      " [-0.42818875 -0.52653139 -0.53655579  0.15429856]\n",
      " [-0.65187788  0.04940777  0.04700627  0.37950998]\n",
      " [-0.31634419 -0.81450097 -0.81591586  0.88968091]]\n"
     ]
    }
   ],
   "source": [
    "A = np.array([[2,1,3,-1],[3,-1,2,0],[1,3,4,-2],[4,-3,1,1]])\n",
    "valeursPropres , vecteursPropres = linalg.eig(A)\n",
    "print(valeursPropres)\n",
    "print(vecteursPropres)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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