{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear Algebra - Exercises"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculate the determinant and inverse of $W$\n",
    "\n",
    "$W=\\begin{bmatrix}1&&2&&-1\\\\1&&0&&1\\\\0&&1&&0\\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(\"det W =\", (1 * (0 - 1)) - (2 * (0 - 0)) + (-1 * (1 - 0)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To calculate the inverse we can leverage the fact that $WW^{-1}=I$.\n",
    "\n",
    "To find $W^{-1}$ we can solve the system of equation formed by the matrix multiplication of $W$ (coefficients) with $W^{-1}$ (variables) and $I$ are the constants on the RHS.\n",
    "\n",
    "Or we can use the augmented matrices method, which basically entails augmenting the matrix $W$ with $I$ and applying row operations so the LHS becomes the reduced echelon form of $W$ and the RHS side becomes $W^{-1}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "W = sp.Matrix([[1, 2, -1], [1, 0, 1], [0, 1, 0]])\n",
    "WI = W.row_join(sp.Matrix.eye(3))\n",
    "WI"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# R2 = R2 - R1\n",
    "WI = WI.elementary_row_op(\"n->n+km\", row=1, k=-1, row1=1, row2=0)\n",
    "WI"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# R2 = -0.5R2\n",
    "WI = WI.elementary_row_op(\"n->kn\", row=1, k=-0.5)\n",
    "WI"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# R1 = R1 - 2R2\n",
    "WI = WI.elementary_row_op(\"n->n+km\", row=0, k=-2, row1=0, row2=1)\n",
    "WI"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# R3 = R3 - R2\n",
    "WI = WI.elementary_row_op(\"n->n+km\", row=2, k=-1, row1=2, row2=1)\n",
    "WI"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# R1 = R1 - R3\n",
    "WI = WI.elementary_row_op(\"n->n+km\", row=0, k=-1, row1=0, row2=2)\n",
    "WI"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# R2 = R2 + R3\n",
    "WI = WI.elementary_row_op(\"n->n+km\", row=1, k=1, row1=1, row2=2)\n",
    "WI"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "WI = WI[-3:, -3:]\n",
    "WI"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can verify $WW^{-1}=I$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "W.multiply(WI)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculate the characteristic polynomial, eigenvalues and eigenvectors of $A$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$A = \\begin{bmatrix}2&&1\\\\-3&&6\\end{bmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$A = \\begin{bmatrix}2-\\lambda&&1\\\\-3&&6-\\lambda\\end{bmatrix}$\n",
    "\n",
    "$det(A-\\lambda I) = (2-\\lambda)(6-\\lambda)-(1)(-3)$\n",
    "\n",
    "$\\lambda^2 - 8\\lambda + 12 + 3$\n",
    "\n",
    "$\\lambda^2 - 8\\lambda + 15$\n",
    "\n",
    "$(5 - \\lambda)(3 -\\lambda)$\n",
    "\n",
    "$\\lambda = 5, \\lambda = 3$\n",
    "\n",
    "For $\\lambda$ = 5:\n",
    "\n",
    "$\\begin{bmatrix}-3&&1\\\\-3&&1\\end{bmatrix}$\n",
    "\n",
    "$\\begin{bmatrix}-3&&1\\\\0&&0\\end{bmatrix}$\n",
    "\n",
    "$\\begin{cases}-3v_1+v_2\\\\0=0\\end{cases}$\n",
    "\n",
    "$\\begin{cases}3v_1=v_2\\\\0=0\\end{cases}$\n",
    "\n",
    "$\\vec{v}_{\\lambda=5} = \\begin{bmatrix}1\\\\3\\end{bmatrix}$\n",
    "\n",
    "For $\\lambda$ = 3:\n",
    "\n",
    "$\\begin{bmatrix}-1&&1\\\\-3&&3\\end{bmatrix}$\n",
    "\n",
    "$\\begin{bmatrix}-1&&1\\\\0&&0\\end{bmatrix}$\n",
    "\n",
    "$\\begin{cases}-v_1+v_2\\\\0=0\\end{cases}$\n",
    "\n",
    "$\\begin{cases}v_1=v_2\\\\0=0\\end{cases}$\n",
    "\n",
    "$\\vec{v}_{\\lambda=3} = \\begin{bmatrix}1\\\\1\\end{bmatrix}$\n"
   ]
  }
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