{
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   "source": [
    "<span style=\"font-size:8pt\">*ENSAM-Bordeaux, Mathématiques et informatique. Date : le 09/11/19. Auteur : Éric Ducasse. Version : 0.9*</span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-danger\"> <span style=\"color:#800000\"> *On pourra faire exécuter ce notebook cellule par cellule $\\left(\\mathtt{Maj+Entrée}\\right)$ ou intégralement par $\\mathtt{\\;Kernel\\rightarrow Restart\\;\\&\\;Run\\;All}$.* </span> </div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import sympy as sb\n",
    "sb.init_printing()\n",
    "print(\"Version de sympy :\",sb.__version__) \n",
    "from IPython.display import display"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <span style=\"color:#0066BB\"> **Calcul formel : TD n°3, troisième partie<br/><br/>$\\hspace{4cm}$*Calcul matriciel* ** </span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-danger\"> <span style=\"color:#800000\"> **Pour chaque exercice, faire exécuter la partie *exemples* cellule par cellule $\\left(\\mathtt{Maj+Entrée}\\right)$, avant de passer à la partie *Travail à faire*.** </span> </div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <span style=\"color: #0000BB\"> *Exercice 18* </span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <span style=\"color:purple\"> 18.1 Objectifs </span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### *<span style=\"color:#005500\"> Savoir définir des matrices, en extraire des sous-matrices, faire les opérations habituelles du calcul matriciel,<br />dont la résolution de systèmes linéaires, et, si une matrice est carrée, calculer exactement son déterminant,<br />sa matrice inverse, ses valeurs propres, ses vecteurs propres. </span>*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <span style=\"color:purple\"> 18.2 Exemples </span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### <span style=\"color:#604000\">Définition d'une matrice</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a,b = sb.symbols(\"a,b\")\n",
    "M = sb.Matrix([[a-b,-b,0,0],[b,a+b,0,0],[0,0,b,0],[0,0,0,b]])\n",
    "M"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### <span style=\"color:#604000\">Dimensions de la matrice</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "M.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### <span style=\"color:#604000\">Extraction d'une ligne (matrice-ligne)</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "R = M.row(2) ; R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "R.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "R == M[2,:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### <span style=\"color:#604000\">Extraction d'une colonne (matrice-colonne)</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "C = M.col(1) ; C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "C.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "C == M[:,1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### <span style=\"color:#604000\">Slicing similaire à celui de $\\mathtt{numpy.ndarray}$</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "M[-3:,:3]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### <span style=\"color:#604000\">Attention :</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "[ M[i] for i in range(16) ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### <span style=\"color:#604000\">Méthodes similaires à celles de $\\mathtt{numpy.ndarray}$</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "M.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "M.trace()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "M.transpose()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "M.transpose() == M.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "P = M.inv() ; P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "P.simplify() ; P # Simplification en place !"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "I = sb.eye(4) ; I"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### <span style=\"color:#A00000\">Attention !</span> l'opérateur $\\mathtt{*}$ n'a pas la même signification pour les objets de type  $\\mathtt{sympy.Matrix}$ et de type $\\mathtt{numpy.ndarray}$. Nous conseillons d'éviter son utilisation et de lui préférer l'opérateur $\\mathtt{@}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(P@M).equals(I)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### <span style=\"color:#604000\">Valeurs et vecteurs propres</span>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "M.eigenvals()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "diago = M.eigenvects() ; diago"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "V0 = diago[0][2][0]\n",
    "V1 = sb.Matrix([[1],[1],[0],[0]]) ; \n",
    "V2 = diago[1][2][0]\n",
    "V3 = diago[1][2][1]\n",
    "LV = [V0,V1,V2,V3] ; LV"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Q = sb.Matrix([ X.T for X in LV ]).T ; Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Qm1 = Q.inv() ; Qm1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Q@M@Qm1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <span style=\"color:purple\"> 18.3 Travail à faire </span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $a)$ Définir la matrice $\\displaystyle A=\\left(\\begin{array}{ccc}\n",
    "1 & a^3 & 0 \\\\ 0 & 1 & b^3 \\\\ c^3 & 0 & 1 \\end{array}\\right)$, où $a$, $b$ et $c$ sont des réels strictement positifs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $b)$ Déterminer son déterminant ainsi que sa matrice inverse."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $c)$ Déterminer les valeurs propres et les vecteurs propres de la matrice $A$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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