LinearAlgebra_GilbertStrang.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "source": [
    "# Linear Algebra by Prof. Gilbert Strang—Notes from a Programmer"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "Using `numpy`, `scipy`, and `matplotlib`, this notebook exhibits the concepts taught through this course:\n",
    "\n",
    "* Gilbert Strang. 18.06 Linear Algebra. Spring 2010. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA. [Direct Link](https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/#)"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "## Contents"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "source": [
    "1. [**Create my chosen array**](<#Create-my-chosen-array>)\n",
    "2. [**Basic operations**](<#Basic-operations>)\n",
    "    * [Transpose](<#Transpose>)\n",
    "        * [Matrix (2D array)](<#Matrix-(2D-array)>)\n",
    "        * [Vector (1D array)](<#Vector-(1D-array)>)\n",
    "        * [Vector (2D array)](<#Vector-(2D-array)>)\n",
    "    * [Inverse](<#Inverse>)\n",
    "3. [**Decompositions**](<#Decompositions>)\n",
    "    * [LU](<#LU>)\n",
    "    * [QR](<#QR>)\n",
    "    * [Diagonalization (Eigendecomposition)](<#Diagonalization-(Eigendecomposition)>)\n",
    "4. [**Subspaces**](<#Subspaces>)\n",
    "    * [Column Space C(A)](<#Column-Space-C(A)>)\n",
    "        * [Manually](<#Manually>)\n",
    "            * [QR](<#QR>)\n",
    "                * [Orthonormal basis](<#Orthonormal-basis>)\n",
    "                * [Independent columns](<#Independent-columns>)\n",
    "            * [LU (Independent columns)](<#LU-(Independent-columns)>)\n",
    "    * [Null Space N(A)](<#Null-Space-N(A)>)\n",
    "    * [Row Space C(At)](<#Row-Space-C(At)>)\n",
    "    * [Left Null Space N(At)](<#Left-Null-Space-N(At)>)\n",
    "5. [**Rank-one matrix**](<#Rank-one-matrix>)\n",
    "6. [**Least Squares**](<#Least-Squares>)\n",
    "    * [Manually](<#Manually>)\n",
    "    * [Fitting a line/parabola](<#Fitting-a-line/parabola>)\n",
    "7. [**Properties of the determinant**](<#Properties-of-the-determinant>)\n",
    "    * [$|AB| = |A||B|$](<#$|AB|-=-|A||B|$>)\n",
    "    * [$|A^{-1}| = \\frac{1}{|A|}$](<#$|A^{-1}|-=-\\frac{1}{|A|}$>)\n",
    "    * [$|A^2|=|A|^2$](<#$|A^2|=|A|^2$>)\n",
    "    * [$|2A|=2^n |A|$](<#$|2A|=2^n-|A|$>)\n",
    "    * [$|A^T|=|A|$](<#$|A^T|=|A|$>)\n",
    "    * [$|U|=|A|$](<#$|U|=|A|$>)\n",
    "    * [$|Q|=1$](<#$|Q|=1$>)\n",
    "8. [**Eigenvalues and eigenvectors**](<#Eigenvalues-and-eigenvectors>)\n",
    "    * [Product of eigenvalues](<#Product-of-eigenvalues>)\n",
    "    * [Sum of eigenvalues](<#Sum-of-eigenvalues>)\n",
    "    * [Adding to the diagonal adds to lambdas](<#Adding-to-the-diagonal-adds-to-lambdas>)\n",
    "    * [Attributes of a diagonal matrix](<#Attributes-of-a-diagonal-matrix>)\n",
    "    * [Rotation matrix often has complex lambdas](<#Rotation-matrix-often-has-complex-lambdas>)\n",
    "9. [**Diagonalizability**](<#Diagonalizability>)\n",
    "    * [Algebraic & Geometric Multiplicity](<#Algebraic-&-Geometric-Multiplicity>)\n",
    "    * [Defective matrix](<#Defective-matrix>)\n",
    "    * [Diagonalizable matrix](<#Diagonalizable-matrix>)\n",
    "        * [Normal](<#Normal>)\n",
    "            * [Hermitian](<#Hermitian>)\n",
    "            * [Unitary](<#Unitary>)\n",
    "10. [**Matrix Power $A^{k}$**](<#Matrix-Power-$A^{k}$>)\n",
    "    * [A diagonal matrix](<#A-diagonal-matrix>)\n",
    "    * [Any matrix](<#Any-matrix>)\n",
    "11. [**Matrix Exponential $e^{At}$**](<#Matrix-Exponential-$e^{At}$>)\n",
    "    * [A diagonal matrix](<#A-diagonal-matrix>)\n",
    "    * [Any matrix](<#Any-matrix>)\n",
    "12. [**Systems of difference equations**](<#Systems-of-difference-equations>)\n",
    "    * [Discrete $u_{k}=A^{k}u_{0}$](<#Discrete-$u_{k}=A^{k}u_{0}$>)\n",
    "        * [e.g. Fibonacci](<#e.g.-Fibonacci>)\n",
    "        * [Convergence](<#Convergence>)\n",
    "            * [Stability](<#Stability>)\n",
    "            * [Steady State](<#Steady-State>)\n",
    "            * [Blow Up](<#Blow-Up>)\n",
    "        * [Analyze](<#Analyze>)\n",
    "            * [Graph](<#Graph>)\n",
    "    * [Continuous $u(t)=e^{At}u(0)$](<#Continuous-$u(t)=e^{At}u(0)$>)\n",
    "        * [e.g. A 2×2 system of diff eqs](<#e.g.-A-2×2-system-of-diff-eqs>)\n",
    "        * [e.g. A 2nd order diff eq](<#e.g.-A-2nd-order-diff-eq>)\n",
    "        * [Convergence](<#Convergence>)\n",
    "            * [Stability](<#Stability>)\n",
    "            * [Steady State](<#Steady-State>)\n",
    "            * [Blow Up](<#Blow-Up>)\n",
    "        * [Analyze](<#Analyze>)\n",
    "            * [Graph](<#Graph>)\n",
    "13. [**Markov matrix**](<#Markov-matrix>)\n",
    "    * [All entries are 0 or positive](<#All-entries-are-0-or-positive>)\n",
    "    * [All columns add to 1](<#All-columns-add-to-1>)\n",
    "    * [$\\exists i: \\lambda_{i} = 1$](<#$\\exists-i:-\\lambda_{i}-=-1$>)\n",
    "    * [$\\forall j \\ne i , |\\lambda_{j}| \\leq 1$](<#$\\forall-j-\\ne-i-,-|\\lambda_{j}|-\\leq-1$>)\n",
    "    * [It converges to a positive steady state](<#It-converges-to-a-positive-steady-state>)\n",
    "    * [$A^{k}$ is still Markov](<#$A^{k}$-is-still-Markov>)\n",
    "14. [**Expansion with a basis**](<#Expansion-with-a-basis>)\n",
    "    * [With A](<#With-A>)\n",
    "    * [With Q (orthonormal basis)](<#With-Q-(orthonormal-basis)>)\n",
    "    * [With S (eigenvectors)](<#With-S-(eigenvectors)>)\n",
    "15. [**Complex Vectors and Matrices**](<#Complex-Vectors-and-Matrices>)\n",
    "    * [Hermitian matrix](<#Hermitian-matrix>)\n",
    "    * [Unitary matrix](<#Unitary-matrix>)\n",
    "16. [**Positive Definite Matrices**](<#Positive-Definite-Matrices>)"
   ],
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  },
  {
   "cell_type": "markdown",
   "source": [
    "## Requirements"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "!pip install matplotlib==3.3.4 numpy==1.20.1 scipy==1.6.0"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n",
     "is_executing": true
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "jupyter": {
     "outputs_hidden": true
    },
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np, numpy.linalg as npLA\n",
    "from numpy.linalg import matrix_power\n",
    "from numpy.testing import assert_allclose, assert_array_less, assert_\n",
    "\n",
    "from scipy import linalg as LA\n",
    "from scipy.linalg import inv, det, expm\n",
    "\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "# Without set_, it can be used as a context manager as well.\n",
    "# formatter={'float_kind':'{:_.1f}'.format}\n",
    "np.set_printoptions(precision=1, suppress=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "[&#x21e7;](#Contents)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "# Create my chosen array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 1]\n",
      " [3 8 1]\n",
      " [0 4 1]]\n"
     ]
    }
   ],
   "source": [
    "def create_a_square_invertible_array():\n",
    "    \"\"\"Invertible (M == N)\"\"\"\n",
    "    return np.array([[1, 2, 1],\n",
    "                     [3, 8, 1],\n",
    "                     [0, 4, 1]])\n",
    "\n",
    "\n",
    "def create_a_square_singular_array():\n",
    "    \"\"\"Singular M == N\"\"\"\n",
    "    return np.array([[1, 1, 4],\n",
    "                     [2, 2, 5],\n",
    "                     [3, 3, 6]])\n",
    "\n",
    "\n",
    "def create_a_Hermitian_array():\n",
    "    \"\"\"Hermitian (M == N)\"\"\"\n",
    "    return np.array([[1, 2j, 1 + 1j],\n",
    "                     [-2j, 8, 4 - 3j],\n",
    "                     [1 - 1j, 4 + 3j, -3]])\n",
    "\n",
    "\n",
    "def create_a_unitary_array():\n",
    "    \"\"\"Unitary (M == N)\"\"\"\n",
    "    return np.array([[.5 + .5j, .5 - .5j],\n",
    "                     [.5 - .5j, .5 + .5j]])\n",
    "\n",
    "\n",
    "def create_a_Markov_matrix():\n",
    "    \"\"\"Markov (M == N)\"\"\"\n",
    "    return np.array([[.1, .01, .3],\n",
    "                     [.2, .99, .3],\n",
    "                     [.7, 0, .4]])\n",
    "\n",
    "\n",
    "def create_a_Jordan_array():\n",
    "    \"\"\"Jordan (M == N)\"\"\"\n",
    "    return np.array([[3, 1, 0],\n",
    "                     [0, 3, 1],\n",
    "                     [0, 0, 3]])\n",
    "\n",
    "\n",
    "def create_a_tall_array():\n",
    "    \"\"\"Tall (M > N)\"\"\"\n",
    "    return np.array([[1, 1],\n",
    "                     [2, 2],\n",
    "                     [3, 4],\n",
    "                     [4, 1]])\n",
    "\n",
    "\n",
    "def create_a_fat_array():\n",
    "    \"\"\"Fat (M < N)\"\"\"\n",
    "    return np.array([[1, 2, 3, 4],\n",
    "                     [1, 2, 4, 1]])\n",
    "\n",
    "\n",
    "def create_a_3d_array_leading_to_stability_in_discrete_diff_eq():\n",
    "    \"\"\"A 3 by 3 array for a discrete differential equation that leads to stability\"\"\"\n",
    "    return np.array([[0.1, 0, 0.5],\n",
    "                     [0, 0.1, 0],\n",
    "                     [-0.5, 0, -0.1]])\n",
    "\n",
    "\n",
    "def create_a_3d_array_leading_to_steady_state_in_discrete_diff_eq():\n",
    "    \"\"\"A 3 by 3 array for a discrete differential equation that leads to steady state\"\"\"\n",
    "    return np.array([[0.2, 0, 1],\n",
    "                     [0, 0.2, 0],\n",
    "                     [-1, 0, -0.1]])\n",
    "\n",
    "\n",
    "def create_a_3d_array_blowing_up_in_discrete_diff_eq():\n",
    "    \"\"\"A 3 by 3 array for a discrete differential equation that blows up\"\"\"\n",
    "    return np.array([[1, 1, 1],\n",
    "                     [-1, 0.5, 0],\n",
    "                     [-1, 0, -0.5]])\n",
    "\n",
    "\n",
    "def create_a_3d_array_leading_to_stability_in_continuous_diff_eq():\n",
    "    \"\"\"A 3 by 3 array for a continuous differential equation that leads to stability\"\"\"\n",
    "    return np.array([[-0.8, 1, 1],\n",
    "                     [-1, -0.6, 1],\n",
    "                     [-1, 1, -0.7]])\n",
    "\n",
    "\n",
    "def create_a_3d_array_leading_to_steady_state_in_continuous_diff_eq():\n",
    "    \"\"\"A 3 by 3 array for a continuous differential equation that leads to steady state\"\"\"\n",
    "    return np.array([[-0.6, 1, 1],\n",
    "                     [-1, -0.5, 1],\n",
    "                     [-1, 1, -0.6]])\n",
    "\n",
    "\n",
    "def create_a_3d_array_blowing_up_in_continuous_diff_eq():\n",
    "    \"\"\"A 3 by 3 array for a continuous differential equation that blows up\"\"\"\n",
    "    return np.array([[1, 1, 1],\n",
    "                     [-1, 0.5, 0],\n",
    "                     [-1, 0, -0.5]])\n",
    "\n",
    "\n",
    "def create_a_2d_array_leading_to_stability_in_discrete_diff_eq():\n",
    "    \"\"\"A 2 by 2 array for a discrete differential equation that leads to stability\"\"\"\n",
    "    return np.array([[0.1, 0.5],\n",
    "                     [-0.5, 0.2]])\n",
    "\n",
    "\n",
    "def create_a_2d_array_leading_to_steady_state_in_discrete_diff_eq():\n",
    "    \"\"\"A 2 by 2 array for a discrete differential equation that leads to steady state\"\"\"\n",
    "    return np.array([[0.8, 0.5],\n",
    "                     [-0.5, 0.9]])\n",
    "\n",
    "\n",
    "def create_a_2d_array_blowing_up_in_discrete_diff_eq():\n",
    "    \"\"\"A 2 by 2 array for a discrete differential equation that blows up\"\"\"\n",
    "    return np.array([[0.5, 1],\n",
    "                     [-1, 0.8]])\n",
    "\n",
    "\n",
    "def create_a_2d_array_leading_to_stability_in_continuous_diff_eq():\n",
    "    \"\"\"A 2 by 2 array for a continuous differential equation that leads to stability\"\"\"\n",
    "    return np.array([[-0.3, 1],\n",
    "                     [-1, -0.1]])\n",
    "\n",
    "\n",
    "def create_a_2d_array_leading_to_steady_state_in_continuous_diff_eq():\n",
    "    \"\"\"A 2 by 2 array for a continuous differential equation that leads to steady state\"\"\"\n",
    "    return np.array([[0, 1],\n",
    "                     [-1, -0.1]])\n",
    "\n",
    "\n",
    "def create_a_2d_array_blowing_up_in_continuous_diff_eq():\n",
    "    \"\"\"A 2 by 2 array for a continuous differential equation that blows up\"\"\"\n",
    "    return np.array([[0.3, 1],\n",
    "                     [-1, 0.1]])\n",
    "\n",
    "\n",
    "def create_my_array():\n",
    "    \"\"\"The array I wish to see its attributes throughout the majority of cells.\n",
    "    Feel free to change it!\"\"\"\n",
    "    return create_a_square_invertible_array()\n",
    "\n",
    "\n",
    "A = create_my_array()\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "[&#x21e7;](#Contents)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "# Basic operations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Transpose"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "### Matrix (2D array)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2]\n",
      " [3 4]]\n"
     ]
    }
   ],
   "source": [
    "A = np.arange(1, 5).reshape((2, 2))\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 3]\n",
      " [2 4]]\n"
     ]
    }
   ],
   "source": [
    "print(A.T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "### Vector (1D array)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 2]\n"
     ]
    }
   ],
   "source": [
    "b = np.arange(3)\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 2]\n"
     ]
    }
   ],
   "source": [
    "print(b.T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "A 1D array is the same as its transpose.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "1D arrays will automatically transform to (perhaps transpose) 2D ones when they are on either side of @ functions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "### Vector (2D array)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0 1 2]]\n"
     ]
    }
   ],
   "source": [
    "x = np.arange(3).reshape((1, 3))\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0]\n",
      " [1]\n",
      " [2]]\n"
     ]
    }
   ],
   "source": [
    "print(x.T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Inverse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.2  0.4]\n",
      " [ 0.6 -0.2]]\n"
     ]
    }
   ],
   "source": [
    "A = np.array([[1, 2],\n",
    "              [3, 1]])\n",
    "Ai = LA.inv(A)\n",
    "print(Ai)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "[&#x21e7;](#Contents)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "# Decompositions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## LU"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 1]\n",
      " [3 8 1]\n",
      " [0 4 1]]\n"
     ]
    }
   ],
   "source": [
    "A = create_my_array()\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "P\n",
      "[[0. 0. 1.]\n",
      " [1. 0. 0.]\n",
      " [0. 1. 0.]]\n",
      "\n",
      "L\n",
      "[[ 1.   0.   0. ]\n",
      " [ 0.   1.   0. ]\n",
      " [ 0.3 -0.2  1. ]]\n",
      "\n",
      "U\n",
      "[[3.  8.  1. ]\n",
      " [0.  4.  1. ]\n",
      " [0.  0.  0.8]]\n"
     ]
    }
   ],
   "source": [
    "P, L, U = LA.lu(A)\n",
    "print('P', P, '', 'L', L, '', 'U', U, sep='\\n')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## QR"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 1]\n",
      " [3 8 1]\n",
      " [0 4 1]]\n"
     ]
    }
   ],
   "source": [
    "A = create_my_array()\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Q\n",
      "[[-0.3  0.1  0.9]\n",
      " [-0.9 -0.  -0.3]\n",
      " [-0.  -1.   0.2]]\n",
      "\n",
      "R\n",
      "[[-3.2 -8.2 -1.3]\n",
      " [ 0.  -4.  -0.9]\n",
      " [ 0.   0.   0.8]]\n"
     ]
    }
   ],
   "source": [
    "Q, R = LA.qr(A)\n",
    "print('Q', Q, '', 'R', R, sep='\\n')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Diagonalization (Eigendecomposition)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 1]\n",
      " [3 8 1]\n",
      " [0 4 1]]\n"
     ]
    }
   ],
   "source": [
    "A = create_my_array()\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "la\n",
      "[9.4+0.j 0.3+1.j 0.3-1.j]\n",
      "\n",
      "S\n",
      "[[ 0.3+0.j   0. -0.5j  0. +0.5j]\n",
      " [ 0.9+0.j  -0.1+0.2j -0.1-0.2j]\n",
      " [ 0.4+0.j   0.8+0.j   0.8-0.j ]]\n"
     ]
    }
   ],
   "source": [
    "la, S = npLA.eig(A)\n",
    "print('la', la, '', 'S', S, sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[9.4+0.j 0. +0.j 0. +0.j]\n",
      " [0. +0.j 0.3+1.j 0. +0.j]\n",
      " [0. +0.j 0. +0.j 0.3-1.j]]\n"
     ]
    }
   ],
   "source": [
    "print(np.diagflat(la))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1.-0.j 2.+0.j 1.+0.j]\n",
      " [3.-0.j 8.-0.j 1.+0.j]\n",
      " [0.-0.j 4.-0.j 1.+0.j]]\n"
     ]
    }
   ],
   "source": [
    "print(S @ np.diagflat(la) @ inv(S))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "[&#x21e7;](#Contents)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "# Subspaces"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 1]\n",
      " [3 8 1]\n",
      " [0 4 1]]\n"
     ]
    }
   ],
   "source": [
    "A = create_my_array()\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Column Space C(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "(Using SVD, Orthonormal basis)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.2  0.1  1. ]\n",
      " [-0.9  0.4 -0.3]\n",
      " [-0.4 -0.9 -0. ]]\n"
     ]
    }
   ],
   "source": [
    "col_sp_A_orthonormal_basis_by_SVD = LA.orth(A)\n",
    "print(col_sp_A_orthonormal_basis_by_SVD)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "### Manually"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "#### QR"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "##### Orthonormal basis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Q\n",
      "[[-0.2 -0.3 -0.9]\n",
      " [-0.9 -0.4  0.3]\n",
      " [-0.4  0.9 -0.2]]\n",
      "\n",
      "R\n",
      "[[-9.2 -2.8 -1.5]\n",
      " [ 0.  -1.4  0.2]\n",
      " [ 0.   0.  -0.8]]\n",
      "\n",
      "P\n",
      "[1 0 2]\n",
      "\n",
      "rank\n",
      "3\n"
     ]
    }
   ],
   "source": [
    "Q, R, P = LA.qr(A, pivoting=True)\n",
    "rank_A = npLA.matrix_rank(A)\n",
    "print('Q', Q, '', 'R', R, '', 'P', P, '', 'rank', rank_A, sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.2 -0.3 -0.9]\n",
      " [-0.9 -0.4  0.3]\n",
      " [-0.4  0.9 -0.2]]\n"
     ]
    }
   ],
   "source": [
    "col_sp_A_orthonormal_basis_by_QR = Q[:, :rank_A]\n",
    "print(col_sp_A_orthonormal_basis_by_QR)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "##### Independent columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Q\n",
      "[[-0.3  0.1  0.9]\n",
      " [-0.9 -0.  -0.3]\n",
      " [-0.  -1.   0.2]]\n",
      "\n",
      "R\n",
      "[[-3.2 -8.2 -1.3]\n",
      " [ 0.  -4.  -0.9]\n",
      " [ 0.   0.   0.8]]\n",
      "\n",
      "rank\n",
      "3\n"
     ]
    }
   ],
   "source": [
    "Q, R = LA.qr(A)\n",
    "print('Q', Q, '', 'R', R, '', 'rank', rank_A, sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def obtain_unique_indices_of_first_nonzero_col_in_each_row(U_resulted_by_decomposition: np.ndarray) -> list:\n",
    "    # I should first convert all the \"-0.\" to \"0.\" so that nonzero() can find them.\n",
    "    U_copy = U_resulted_by_decomposition.copy()\n",
    "    U_copy[abs(U_copy) < 1.e-7] = 0\n",
    "\n",
    "    # Because some rows in U may not have even one nonzero element,\n",
    "    # I have to find the index for the first one in two steps.\n",
    "    index_of_all_nonzero_cols_in_each_row = (\n",
    "        [U_copy[i, :].nonzero()[0] for i in range(U_copy.shape[0])]\n",
    "    )\n",
    "\n",
    "    index_of_first_nonzero_col_in_each_row = (\n",
    "        [indices[0] for indices in index_of_all_nonzero_cols_in_each_row\n",
    "         if len(indices) > 0]\n",
    "    )\n",
    "\n",
    "    # Because two rows or more may have the same indices for their first nonzero element,\n",
    "    # I should remove duplicates.\n",
    "    unique_indices = sorted(list(set(index_of_first_nonzero_col_in_each_row)))\n",
    "    return unique_indices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 1]\n",
      " [3 8 1]\n",
      " [0 4 1]]\n"
     ]
    }
   ],
   "source": [
    "index_of_linearly_independent_cols_of_A = obtain_unique_indices_of_first_nonzero_col_in_each_row(R)\n",
    "\n",
    "col_sp_A_independent_columns_by_QR = A[:, index_of_linearly_independent_cols_of_A]\n",
    "\n",
    "print(col_sp_A_independent_columns_by_QR)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "We can create them by Q and R too:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1. 2. 1.]\n",
      " [3. 8. 1.]\n",
      " [0. 4. 1.]]\n"
     ]
    }
   ],
   "source": [
    "col_sp_A_independent_columns_by_QR = Q @ R[:, index_of_linearly_independent_cols_of_A]\n",
    "\n",
    "print(col_sp_A_independent_columns_by_QR)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "#### LU (Independent columns)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "P\n",
      "[[0. 0. 1.]\n",
      " [1. 0. 0.]\n",
      " [0. 1. 0.]]\n",
      "\n",
      "L\n",
      "[[ 1.   0.   0. ]\n",
      " [ 0.   1.   0. ]\n",
      " [ 0.3 -0.2  1. ]]\n",
      "\n",
      "U\n",
      "[[3.  8.  1. ]\n",
      " [0.  4.  1. ]\n",
      " [0.  0.  0.8]]\n"
     ]
    }
   ],
   "source": [
    "P, L, U = LA.lu(A)\n",
    "print('P', P, '', 'L', L, '', 'U', U, sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 1]\n",
      " [3 8 1]\n",
      " [0 4 1]]\n"
     ]
    }
   ],
   "source": [
    "index_of_linearly_independent_cols_of_A = obtain_unique_indices_of_first_nonzero_col_in_each_row(U)\n",
    "\n",
    "col_sp_A_independent_columns_by_LU = A[:, index_of_linearly_independent_cols_of_A]\n",
    "\n",
    "print(col_sp_A_independent_columns_by_LU)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "We can create them by P, L, and U too:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1. 2. 1.]\n",
      " [3. 8. 1.]\n",
      " [0. 4. 1.]]\n"
     ]
    }
   ],
   "source": [
    "col_sp_A_independent_columns_by_LU = P @ L @ U[:, index_of_linearly_independent_cols_of_A]\n",
    "\n",
    "print(col_sp_A_independent_columns_by_LU)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Null Space N(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "(Using SVD, Orthonormal basis)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[]\n"
     ]
    }
   ],
   "source": [
    "null_sp_A = LA.null_space(A)\n",
    "print(null_sp_A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "0.0"
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.norm(null_sp_A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Row Space C(At)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.3  0.9  0.3]\n",
      " [-0.9 -0.2 -0.2]\n",
      " [-0.2 -0.3  0.9]]\n"
     ]
    }
   ],
   "source": [
    "row_sp_A = LA.orth(A.T)\n",
    "print(row_sp_A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Left Null Space N(At)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[]\n"
     ]
    }
   ],
   "source": [
    "left_null_sp_A = LA.null_space(A.T)\n",
    "print(left_null_sp_A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "[&#x21e7;](#Contents)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "\n",
    "# Rank-one matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1  4  5]\n",
      " [ 2  8 10]\n",
      " [ 3 12 15]]\n"
     ]
    }
   ],
   "source": [
    "u = np.array([[1],\n",
    "              [2],\n",
    "              [3]])\n",
    "\n",
    "v = np.array([[1],\n",
    "              [4],\n",
    "              [5]])\n",
    "\n",
    "A = u @ v.T\n",
    "\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "1"
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "npLA.matrix_rank(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "Why is computing determinant and eigenvalues for rank-one matrices **easy**?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "All eigenvalues of a rank-one matrix are _zeros_ except one of them, which happens when `u` is not orthogonal to `v` (i.e. when `A` is diagonalizable) and is given by a simple dot product:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[24]]\n"
     ]
    }
   ],
   "source": [
    "print(u.T @ v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[24.+0.j -0.+0.j  0.+0.j]\n"
     ]
    }
   ],
   "source": [
    "print(LA.eigvals(A))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "Because rank-one matrices are _singular_, their determinant is simply _zero_."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "0.0"
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LA.det(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "If I write a matrix as a sum of only rank-one matrices, how can it help with computing eigenvalues for the original matrix?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "See the good answers for this question on _math.stackexchange.com_: [Eigenvalues of a sum of rank-one matrices?](https://math.stackexchange.com/q/112186/890001)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "[&#x21e7;](#Contents)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "# Least Squares"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A\n",
      "[[1 1]\n",
      " [1 2]\n",
      " [1 3]]\n",
      "\n",
      "b\n",
      "[1 2 2]\n"
     ]
    }
   ],
   "source": [
    "A = np.column_stack([np.ones(3), np.arange(1, 4)]).astype('int')\n",
    "b = np.array([1, 2, 2])\n",
    "print('A', A, '', 'b', b, sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.7 0.5]\n"
     ]
    }
   ],
   "source": [
    "x_hat, *_ = LA.lstsq(A, b)\n",
    "print(x_hat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "The power of `lstsq()` is that it does not need N(A) == 0."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Manually"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "In this manual approach, all columns of A should be linearly independent."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "At_A\n",
      "[[ 3  6]\n",
      " [ 6 14]]\n",
      "\n",
      "At_b\n",
      "[ 5 11]\n"
     ]
    }
   ],
   "source": [
    "At_A = A.T @ A\n",
    "At_b = A.T @ b\n",
    "print('At_A', At_A, '', 'At_b', At_b, sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.7 0.5]\n"
     ]
    }
   ],
   "source": [
    "x_hat = inv(At_A) @ At_b\n",
    "print(x_hat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "Or:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.7 0.5]\n"
     ]
    }
   ],
   "source": [
    "x_hat = LA.solve(At_A, At_b)\n",
    "print(x_hat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Fitting a line/parabola"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Residues\n",
      "line:\t\t3.1152232142857175\n",
      "parabola:\t0.3954624919193907\n"
     ]
    }
   ],
   "source": [
    "pt_x = np.array([1., 2.5, 3.5, 4., 5., 7., 8.5])\n",
    "pt_y = np.array([0.3, 1.1, 1.5, 2.0, 3.2, 6.6, 8.6])\n",
    "\n",
    "line_A = np.column_stack([pt_x, np.ones(len(pt_x))])\n",
    "line_x_hat, line_residues, *_ = LA.lstsq(line_A, pt_y)\n",
    "line_a, line_b = line_x_hat\n",
    "\n",
    "parabola_A = np.column_stack([pt_x ** 2, pt_x, np.ones(len(pt_x))])\n",
    "parabola_x_hat, parabola_residues, *_ = LA.lstsq(parabola_A, pt_y)\n",
    "parabola_a, parabola_b, parabola_c = parabola_x_hat\n",
    "\n",
    "print('Residues')\n",
    "print('line:\\t', line_residues, sep='\\t')\n",
    "print('parabola:', parabola_residues, sep='\\t')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "<matplotlib.legend.Legend at 0x121e7fe50>"
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": "<Figure size 432x288 with 1 Axes>",
      "image/png": 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\n"
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(pt_x, pt_y, 'o', label='Original data')\n",
    "\n",
    "input_x = np.arange(0, 9, .1)\n",
    "plt.plot(input_x, line_a * input_x + line_b, label='Fitted line')\n",
    "plt.plot(input_x, parabola_a * input_x ** 2 + parabola_b * input_x + parabola_c, label='Fitted parabola')\n",
    "plt.grid(alpha=0.25)\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "[&#x21e7;](#Contents)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "# Properties of the determinant"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "A, B = [np.random.rand(3, 3) for _ in range(2)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## $|AB| = |A||B|$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "assert_allclose(\n",
    "    det(A @ B),\n",
    "    det(A) * det(B)\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## $|A^{-1}| = \\frac{1}{|A|}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "assert_allclose(\n",
    "    det(inv(A)),\n",
    "    1 / det(A)\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## $|A^2|=|A|^2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "assert_allclose(\n",
    "    det(matrix_power(A, 2)),\n",
    "    det(A) ** 2\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## $|2A|=2^n |A|$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "assert_allclose(\n",
    "    det(2 * A),\n",
    "    2 ** A.shape[0] * det(A)\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## $|A^T|=|A|$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "assert_allclose(\n",
    "    det(A.T),\n",
    "    det(A)\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## $|U|=|A|$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "*_, U = LA.lu(A)\n",
    "assert_allclose(\n",
    "    abs(det(U)),\n",
    "    abs(det(A)),\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## $|Q|=1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "Q_a_unitary_matrix = LA.orth(A)\n",
    "assert_allclose(\n",
    "    abs(det(Q_a_unitary_matrix)),\n",
    "    1,\n",
    "    atol=0.1\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "[&#x21e7;](#Contents)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "# Eigenvalues and eigenvectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 1]\n",
      " [3 8 1]\n",
      " [0 4 1]]\n"
     ]
    }
   ],
   "source": [
    "A = create_my_array()\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "la\n",
      "[9.4+0.j 0.3+1.j 0.3-1.j]\n",
      "\n",
      "S\n",
      "[[ 0.3+0.j   0. -0.5j  0. +0.5j]\n",
      " [ 0.9+0.j  -0.1+0.2j -0.1-0.2j]\n",
      " [ 0.4+0.j   0.8+0.j   0.8-0.j ]]\n"
     ]
    }
   ],
   "source": [
    "la, S = npLA.eig(A)\n",
    "print('la', la, '', 'S', S, sep='\\n')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Product of eigenvalues"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "from functools import reduce\n",
    "from operator import mul\n",
    "\n",
    "product_of_eigenvalues = reduce(mul, la)\n",
    "assert_allclose(\n",
    "    product_of_eigenvalues,\n",
    "    det(A)\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Sum of eigenvalues"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "10"
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.trace()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "sum_of_eigenvalues = la.sum()\n",
    "assert_allclose(\n",
    "    sum_of_eigenvalues,\n",
    "    A.trace()\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Adding to the diagonal adds to lambdas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "assert_allclose(\n",
    "    LA.eigvals(A + 100 * np.identity(3)),\n",
    "    la + 100\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Attributes of a diagonal matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 0 0]\n",
      " [0 2 0]\n",
      " [0 0 3]]\n"
     ]
    }
   ],
   "source": [
    "D = np.diagflat(np.arange(1, 4))\n",
    "print(D)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "la\n",
      "[1.+0.j 2.+0.j 3.+0.j]\n",
      "\n",
      "S\n",
      "[[1. 0. 0.]\n",
      " [0. 1. 0.]\n",
      " [0. 0. 1.]]\n"
     ]
    }
   ],
   "source": [
    "la, S = LA.eig(D)\n",
    "print('la', la, '', 'S', S, sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def test_in_a_diagonal_matrix_eigenvalues_are_the_ones_on_the_diagonal():\n",
    "    assert_allclose(\n",
    "        np.diag(D),\n",
    "        la\n",
    "    )\n",
    "\n",
    "\n",
    "test_in_a_diagonal_matrix_eigenvalues_are_the_ones_on_the_diagonal()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def test_in_a_diagonal_matrix_eigenvectors_form_the_identity():\n",
    "    assert_allclose(\n",
    "        S,\n",
    "        np.identity(D.shape[0])\n",
    "    )\n",
    "\n",
    "\n",
    "test_in_a_diagonal_matrix_eigenvectors_form_the_identity()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Rotation matrix often has complex lambdas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0. -1.]\n",
      " [ 1.  0.]]\n"
     ]
    }
   ],
   "source": [
    "from numpy import sin, cos\n",
    "\n",
    "\n",
    "def generate_2_by_2_rotation_matrix(theta):\n",
    "    return np.array([[cos(theta), -sin(theta)],\n",
    "                     [sin(theta), cos(theta)]])\n",
    "\n",
    "\n",
    "R_90 = generate_2_by_2_rotation_matrix(np.pi / 2)\n",
    "print(R_90)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def test_R_90_is_antisymmetric():\n",
    "    assert_allclose(\n",
    "        -R_90.T,\n",
    "        R_90,\n",
    "        atol=1e-7\n",
    "    )\n",
    "\n",
    "\n",
    "test_R_90_is_antisymmetric()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.+1.j 0.-1.j]\n"
     ]
    }
   ],
   "source": [
    "la = LA.eigvals(R_90)\n",
    "print(la)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def test_R_90_has_pure_imaginary_lambdas():\n",
    "    assert_allclose(\n",
    "        la.real,\n",
    "        np.zeros_like(la.real),\n",
    "        atol=1e-7\n",
    "    )\n",
    "\n",
    "\n",
    "test_R_90_has_pure_imaginary_lambdas()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "[&#x21e7;](#Contents)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    },
    "tags": []
   },
   "source": [
    "# Diagonalizability"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Algebraic & Geometric Multiplicity"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 1]\n",
      " [3 8 1]\n",
      " [0 4 1]]\n"
     ]
    }
   ],
   "source": [
    "A = create_my_array()\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "I've used the `eig()` by `numpy.linalg` because it doesn't show _zero_ imaginary parts."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "la\n",
      "[9.4+0.j 0.3+1.j 0.3-1.j]\n",
      "\n",
      "S\n",
      "[[ 0.3+0.j   0. -0.5j  0. +0.5j]\n",
      " [ 0.9+0.j  -0.1+0.2j -0.1-0.2j]\n",
      " [ 0.4+0.j   0.8+0.j   0.8-0.j ]]\n"
     ]
    }
   ],
   "source": [
    "la, S = npLA.eig(A)\n",
    "print('la', la, '', 'S', S, sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "lambda     A.M   G.M\n",
      "9.4+0.0j     1     1\n",
      "0.3+1.0j     1     1\n",
      "0.3-1.0j     1     1\n"
     ]
    }
   ],
   "source": [
    "from collections import Counter\n",
    "\n",
    "\n",
    "def compute_algebraic_multiplicities(M: np.ndarray) -> dict:\n",
    "    return Counter(npLA.eigvals(M))\n",
    "\n",
    "\n",
    "def compute_geometric_multiplicities(M: np.ndarray) -> dict:\n",
    "    geo_multiplicities = {}\n",
    "\n",
    "    for l in npLA.eigvals(M):\n",
    "        A_minus_lambda_I = M - l * np.identity(M.shape[0])\n",
    "        eigenspace = LA.null_space(A_minus_lambda_I)\n",
    "        dim_of_eigenspace = eigenspace.shape[1]\n",
    "\n",
    "        geo_multiplicities[l] = dim_of_eigenspace\n",
    "\n",
    "    return geo_multiplicities\n",
    "\n",
    "\n",
    "def print_alg_and_geo_multiplicities(M: np.ndarray):\n",
    "    alg_ms = compute_algebraic_multiplicities(M)\n",
    "    geo_ms = compute_geometric_multiplicities(M)\n",
    "\n",
    "    print('lambda  ', 'A.M', 'G.M', sep='   ')\n",
    "\n",
    "    for l in sorted(alg_ms, reverse=True):\n",
    "        print(f'{l:#8.1f}', f'{alg_ms[l]:#3d}', f'{geo_ms[l]:#3d}', sep='   ')\n",
    "\n",
    "\n",
    "print_alg_and_geo_multiplicities(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Defective matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "Any nontrivial Jordan block of 2×2 or larger (that is, not completely diagonal) is defective (degenerate)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[3 1]\n",
      " [0 3]]\n"
     ]
    }
   ],
   "source": [
    "J = np.array([[3, 1],\n",
    "              [0, 3]])\n",
    "print(J)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "la\n",
      "[3.+0.j 3.+0.j]\n",
      "\n",
      "S\n",
      "[[ 1. -1.]\n",
      " [ 0.  0.]]\n"
     ]
    }
   ],
   "source": [
    "la, S = LA.eig(J)\n",
    "print('la', la, '', 'S', S, sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "assert_allclose(\n",
    "    npLA.matrix_rank(S),\n",
    "    1,\n",
    "    atol=0.1\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "This Jordan matrix has only one eigenvector."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Diagonalizable matrix\n"
   ]
  },
  {
   "cell_type": "markdown",
   "source": [
    "$A=S \\Lambda S^{-1}$"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "### Normal"
   ]
  },
  {
   "cell_type": "markdown",
   "source": [
    "The square matrix A is _normal_ $\\Longleftrightarrow$ $AA^H = A^HA$."
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "A _normal_ matrix is never defective. (It is always _diagonalizable_.)"
   ]
  },
  {
   "cell_type": "markdown",
   "source": [
    "Matrices are diagonalizable by _unitary_ matrices ($A=Q\\Lambda Q^H$) if and only if they are _normal_."
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "#### Hermitian"
   ]
  },
  {
   "cell_type": "markdown",
   "source": [
    "A Hermitian matrix is always _normal_, so it is always _diagonalizable_ by _unitary_ matrices."
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "#### Unitary"
   ]
  },
  {
   "cell_type": "markdown",
   "source": [
    "A Unitary matrix is always _normal_."
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "[&#x21e7;](#Contents)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "# Matrix Power $A^{k}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## A diagonal matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 0 0]\n",
      " [0 2 0]\n",
      " [0 0 3]]\n"
     ]
    }
   ],
   "source": [
    "D = np.diagflat(np.arange(1, 4))\n",
    "print(D)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 0 0]\n",
      " [0 4 0]\n",
      " [0 0 9]]\n"
     ]
    }
   ],
   "source": [
    "k = 2\n",
    "print(matrix_power(D, k))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "assert_allclose(\n",
    "    matrix_power(D, k),\n",
    "    D ** k\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Any matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 1]\n",
      " [3 8 1]\n",
      " [0 4 1]]\n"
     ]
    }
   ],
   "source": [
    "A = create_my_array()\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "If A is diagonalizable, $A^k=SΛ^kS^{-1}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "la\n",
      "[9.4+0.j 0.3+1.j 0.3-1.j]\n",
      "\n",
      "S\n",
      "[[ 0.3+0.j   0. -0.5j  0. +0.5j]\n",
      " [ 0.9+0.j  -0.1+0.2j -0.1-0.2j]\n",
      " [ 0.4+0.j   0.8+0.j   0.8-0.j ]]\n"
     ]
    }
   ],
   "source": [
    "la, S = LA.eig(A)\n",
    "print('la', la, '', 'S', S, sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "k = 2\n",
    "\n",
    "assert_allclose(\n",
    "    matrix_power(A, k),\n",
    "    S @ np.diagflat(la) ** k @ inv(S)\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "[&#x21e7;](#Contents)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "# Matrix Exponential $e^{At}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## A diagonal matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 0 0]\n",
      " [0 2 0]\n",
      " [0 0 3]]\n"
     ]
    }
   ],
   "source": [
    "D = np.diagflat(np.arange(1, 4))\n",
    "print(D)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 2.7  0.   0. ]\n",
      " [ 0.   7.4  0. ]\n",
      " [ 0.   0.  20.1]]\n"
     ]
    }
   ],
   "source": [
    "print(expm(D))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "t = 2\n",
    "\n",
    "assert_allclose(\n",
    "    expm(D * t),\n",
    "    np.diagflat(np.e ** (np.diag(D) * t))\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Any matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 1]\n",
      " [3 8 1]\n",
      " [0 4 1]]\n"
     ]
    }
   ],
   "source": [
    "A = create_my_array()\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "If A is diagonalizable, $e^{At}=Se^{Λt}S^{-1}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "la\n",
      "[9.4+0.j 0.3+1.j 0.3-1.j]\n",
      "\n",
      "S\n",
      "[[ 0.3+0.j   0. -0.5j  0. +0.5j]\n",
      " [ 0.9+0.j  -0.1+0.2j -0.1-0.2j]\n",
      " [ 0.4+0.j   0.8+0.j   0.8-0.j ]]\n"
     ]
    }
   ],
   "source": [
    "la, S = LA.eig(A)\n",
    "print('la', la, '', 'S', S, sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "t = 2\n",
    "\n",
    "assert_allclose(\n",
    "    expm(A * t),\n",
    "    S @ expm(np.diagflat(la) * t) @ inv(S)\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "[&#x21e7;](#Contents)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "# Systems of difference equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Discrete $u_{k}=A^{k}u_{0}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "### e.g. Fibonacci"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "\n",
    "$f_{0}$ | $f_{1}$ | $f_{2}$ | $f_{3}$ | $f_{4}$ | $f_{5}$ | $f_{6}$ | $f_{7}$ | $...$ | $f_{100}$\n",
    ":- | :- | :- | :- | :- | :- | :- | :- | :- | :-\n",
    "0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | $...$ | ?\n",
    "\n",
    "\n",
    "$f_{k+2} = f_{k+1} + f_{k} \\ \\rightarrow \\ $ 2nd order differential equation (with 2nd derivative)\n",
    "\n",
    "\"I want to get a two by two system first order, instead of a scalar system second order.\"\n",
    "\n",
    "1st order system\n",
    "$\n",
    "\\ \\rightarrow \\\n",
    "\\begin{cases}\n",
    "f_{k+1} + f_{k} = f_{k+2} \\\\ \n",
    "f_{k+1} + 0 = f_{k+1}\n",
    "\\end{cases}\n",
    "\\ \\rightarrow \\\n",
    "\\begin{bmatrix}\n",
    "1 & 1 \\\\\n",
    "1 & 0\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "f_{k+1} \\\\\n",
    "f_{k}\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "f_{k+2} \\\\\n",
    "f_{k+1}\n",
    "\\end{bmatrix}\n",
    "\\ \\rightarrow \\\n",
    "\\begin{bmatrix}\n",
    "1 & 1 \\\\\n",
    "1 & 0\n",
    "\\end{bmatrix}\n",
    "u_{k} = u_{k+1}\n",
    "$\n",
    "\n",
    "$\n",
    "\\Rightarrow\n",
    "u_{k+1} = A u_{k}\n",
    "\\ , \\\n",
    "u_{0} =\n",
    "\\begin{bmatrix}\n",
    "f_{1} \\\\\n",
    "f_{0}\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "1 \\\\\n",
    "0\n",
    "\\end{bmatrix}\n",
    "$\n",
    "\n",
    "$\n",
    "\\Rightarrow\n",
    "u_{k} = A^{k} u_{0} = S \\Lambda^{k} c \\ , \\ c = S^{-1} u_{0}\n",
    "$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "A = np.array([[1, 1],\n",
    "              [1, 0]])\n",
    "\n",
    "u_0 = np.array([1, 0], dtype='float')\n",
    "\n",
    "\n",
    "def u_by_recursion(k_):\n",
    "    global A, u_0\n",
    "\n",
    "    if k_ == 0:\n",
    "        return u_0\n",
    "    else:\n",
    "        return A @ u_by_recursion(k_ - 1)\n",
    "\n",
    "\n",
    "def u_by_diagonalization(k_):\n",
    "    global A, u_0\n",
    "\n",
    "    la_, S_ = npLA.eig(A)\n",
    "\n",
    "    if k_ == 0:\n",
    "        return u_0\n",
    "    else:\n",
    "        return S_ @ np.diagflat(la_ ** k_) @ inv(S_) @ u_0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def assert_diff_fib_are_close():\n",
    "    n = 100\n",
    "    assert_allclose(\n",
    "        u_by_recursion(n),\n",
    "        u_by_diagonalization(n)\n",
    "    )\n",
    "\n",
    "\n",
    "assert_diff_fib_are_close()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "[0, 1, 1, 2, 3, 5, 8, 13]"
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[round(u_by_diagonalization(n)[1]) for n in range(8)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "354_224_848_179_263_045_632\n"
     ]
    }
   ],
   "source": [
    "print(f'{round(u_by_diagonalization(100)[1]):_}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "### Convergence"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "#### Stability"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "$\\forall i: |\\lambda_{i}|<1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def matrix_power_leads_to_stability(la_):\n",
    "    return (np.absolute(la_) < .9).all()\n",
    "\n",
    "\n",
    "tolerance_factor = 1_000_000\n",
    "\n",
    "\n",
    "def assert_norm_of_last_vector_is_by_far_smaller_than_first_one(v_0, v_last):\n",
    "    assert_array_less(\n",
    "        tolerance_factor * LA.norm(v_last),\n",
    "        LA.norm(v_0)\n",
    "    )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "#### Steady State"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "$\\exists i: |\\lambda_{i}|=1, \\forall j: |\\lambda_{j}| \\leq 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def matrix_power_leads_to_steady_state(la_):\n",
    "    return (np.absolute(la_) > .9).any() and (np.absolute(la_) <= 1.1).all()\n",
    "\n",
    "\n",
    "def assert_norm_of_last_vector_is_not_far_away_than_first_one(v_0, v_last):\n",
    "    # It doesn't blow up.\n",
    "    assert_array_less(\n",
    "        LA.norm(v_last),\n",
    "        tolerance_factor * LA.norm(v_0)\n",
    "    )\n",
    "\n",
    "    # It doesn't lead to stability.\n",
    "    assert_array_less(\n",
    "        LA.norm(v_0),\n",
    "        tolerance_factor * LA.norm(v_last)\n",
    "    )\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "#### Blow Up"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "$\\exists i : |\\lambda_{i}|>1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def matrix_power_blows_up(la_):\n",
    "    return (np.absolute(la_) > 1.1).any()\n",
    "\n",
    "\n",
    "def assert_norm_of_last_vector_is_by_far_bigger_than_first_one(v_0, v_last):\n",
    "    try:\n",
    "        assert_(np.isinf(v_last).any())\n",
    "\n",
    "    except AssertionError:\n",
    "        assert_array_less(\n",
    "            tolerance_factor * LA.norm(v_0),\n",
    "            LA.norm(v_last)\n",
    "        )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "### Analyze"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A\n",
      "[[1 2 1]\n",
      " [3 8 1]\n",
      " [0 4 1]]\n",
      "\n",
      "u_0\n",
      "[[1.]\n",
      " [1.]\n",
      " [1.]]\n"
     ]
    }
   ],
   "source": [
    "A = create_my_array()\n",
    "\n",
    "u_0 = np.ones(A.shape[0])\n",
    "\n",
    "print('A', A, '', 'u_0', u_0[:, np.newaxis], sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "la\n",
      "[9.4+0.j 0.3+1.j 0.3-1.j]\n",
      "\n",
      "la_abs\n",
      "[9.4 1.  1. ]\n"
     ]
    }
   ],
   "source": [
    "la = npLA.eigvals(A)\n",
    "print('la', la, '', sep='\\n')\n",
    "\n",
    "print('la_abs', np.absolute(la), sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[91mIt blows up!\u001B[0m\n",
      "\n",
      "u_k\n",
      "[[ 1.7e+18]\n",
      " [-1.3e+19]\n",
      " [-3.4e+18]]\n"
     ]
    }
   ],
   "source": [
    "k = 100\n",
    "A_to_the_k = matrix_power(A, k)\n",
    "u_k = A_to_the_k @ u_0\n",
    "\n",
    "\n",
    "class Styles:\n",
    "    PURPLE = '\\033[95m'\n",
    "    CYAN = '\\033[96m'\n",
    "    DARK_CYAN = '\\033[36m'\n",
    "    BLUE = '\\033[94m'\n",
    "    GREEN = '\\033[92m'\n",
    "    YELLOW = '\\033[93m'\n",
    "    RED = '\\033[91m'\n",
    "    BOLD = '\\033[1m'\n",
    "    UNDERLINE = '\\033[4m'\n",
    "    END = '\\033[0m'\n",
    "\n",
    "\n",
    "convergence_message = {\n",
    "    'stability': f'{Styles.GREEN}It is stable!{Styles.END}',\n",
    "    'steady_state': f'{Styles.YELLOW}It approaches a steady state!{Styles.END}',\n",
    "    'blow_up': f'{Styles.RED}It blows up!{Styles.END}'\n",
    "}\n",
    "\n",
    "convergence_result = ''\n",
    "\n",
    "if matrix_power_leads_to_stability(la):\n",
    "    assert_norm_of_last_vector_is_by_far_smaller_than_first_one(u_0, u_k)\n",
    "\n",
    "    convergence_result = 'stability'\n",
    "\n",
    "if matrix_power_leads_to_steady_state(la):\n",
    "    assert_norm_of_last_vector_is_not_far_away_than_first_one(u_0, u_k)\n",
    "\n",
    "    convergence_result = 'steady_state'\n",
    "\n",
    "if matrix_power_blows_up(la):\n",
    "    assert_norm_of_last_vector_is_by_far_bigger_than_first_one(u_0, u_k)\n",
    "\n",
    "    convergence_result = 'blow_up'\n",
    "\n",
    "print(convergence_message[convergence_result])\n",
    "\n",
    "# print('', 'A_to_the_k', A_to_the_k, sep='\\n')\n",
    "\n",
    "print('', 'u_k', u_k[:, np.newaxis], sep='\\n')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "#### Graph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "<Figure size 504x504 with 1 Axes>",
      "image/png": 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\n"
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def draw_2d_convergence_graph(x_, y_, color):\n",
    "    fig, ax = plt.subplots(figsize=(7, 7))\n",
    "\n",
    "    ax.plot(x_, y_, 'o-', lw=0.5, color=color)\n",
    "\n",
    "    # Compute max_lim based on plotted data\n",
    "    x_lim = abs(max(ax.get_xlim(), key=abs))\n",
    "    y_lim = abs(max(ax.get_ylim(), key=abs))\n",
    "    max_lim = max(x_lim, y_lim)\n",
    "\n",
    "    # Position xyz axes to the center\n",
    "    ax.set_xlim(xmin=-max_lim, xmax=max_lim)\n",
    "    ax.set_ylim(ymin=-max_lim, ymax=max_lim)\n",
    "\n",
    "    ax.grid(True, lw=.5)\n",
    "\n",
    "\n",
    "from matplotlib.patches import FancyArrowPatch\n",
    "from mpl_toolkits.mplot3d import proj3d\n",
    "\n",
    "\n",
    "class Arrow3D(FancyArrowPatch):\n",
    "    \"\"\"\n",
    "    Ref: https://stackoverflow.com/a/22867877/4526384\n",
    "    \"\"\"\n",
    "\n",
    "    def __init__(self, xs, ys, zs, *args, **kwargs):\n",
    "        FancyArrowPatch.__init__(self, (0, 0), (0, 0), *args, **kwargs)\n",
    "        self._verts3d = xs, ys, zs\n",
    "\n",
    "    def draw(self, renderer):\n",
    "        xs3d, ys3d, zs3d = self._verts3d\n",
    "        xs, ys, zs = proj3d.proj_transform(xs3d, ys3d, zs3d, renderer.M)\n",
    "        self.set_positions((xs[0], ys[0]), (xs[1], ys[1]))\n",
    "        FancyArrowPatch.draw(self, renderer)\n",
    "\n",
    "\n",
    "def draw_xyz_axes_at_center(mpl_ax):\n",
    "    # Compute max_lim based on plotted data\n",
    "    x_lim = abs(max(mpl_ax.get_xlim(), key=abs))\n",
    "    y_lim = abs(max(mpl_ax.get_ylim(), key=abs))\n",
    "    z_lim = abs(max(mpl_ax.get_zlim(), key=abs))\n",
    "    max_lim = max(x_lim, y_lim, z_lim)\n",
    "\n",
    "    # Position xyz axes at the center\n",
    "    mpl_ax.set_xlim(-max_lim, max_lim)\n",
    "    mpl_ax.set_ylim(-max_lim, max_lim)\n",
    "    mpl_ax.set_zlim(-max_lim, max_lim)\n",
    "\n",
    "    # Draw xyz axes\n",
    "    axes = ['x', 'y', 'z']\n",
    "    for i, axis in enumerate(axes):\n",
    "        start_end_pts = np.zeros((3, 2))\n",
    "        start_end_pts[i] = [-max_lim, max_lim]\n",
    "\n",
    "        # Draw axis\n",
    "        xs, ys, zs = start_end_pts[0], start_end_pts[1], start_end_pts[2]\n",
    "\n",
    "        a = Arrow3D(xs, ys, zs,\n",
    "                    mutation_scale=20, arrowstyle='-|>', color='black')\n",
    "        mpl_ax.add_artist(a)\n",
    "\n",
    "        # Add label\n",
    "        end_pt_with_padding = start_end_pts[:, 1] * 1.1\n",
    "\n",
    "        mpl_ax.text(*end_pt_with_padding,\n",
    "                    axis,\n",
    "                    horizontalalignment='center',\n",
    "                    verticalalignment='center',\n",
    "                    color='black')\n",
    "\n",
    "\n",
    "def draw_3d_convergence_graph(x_, y_, z_, color):\n",
    "    ax = plt.figure(figsize=(7, 7)).add_subplot(projection='3d')\n",
    "\n",
    "    ax.plot(x_, y_, z_, '.-', lw=0.5, color=color)\n",
    "\n",
    "    draw_xyz_axes_at_center(ax)\n",
    "\n",
    "    ax.set_xlabel('x axis')\n",
    "    ax.set_ylabel('y axis')\n",
    "    ax.set_zlabel('z axis')\n",
    "\n",
    "    plt.show()\n",
    "\n",
    "\n",
    "def draw_convergence_graph(u_, color):\n",
    "    if u_.shape[0] == 2:\n",
    "        draw_2d_convergence_graph(u_[0, :], u_[1, :], color)\n",
    "\n",
    "    elif u_.shape[0] == 3:\n",
    "        draw_3d_convergence_graph(u_[0, :], u_[1, :], u_[2, :], color)\n",
    "\n",
    "\n",
    "graph_color = {\n",
    "    'stability': 'green',\n",
    "    'steady_state': 'yellow',\n",
    "    'blow_up': 'red'\n",
    "}\n",
    "u = [u_0]\n",
    "\n",
    "for i in range(k):\n",
    "    u.append(A @ u[-1])\n",
    "\n",
    "# %matplotlib notebook\n",
    "\n",
    "u = np.array(u).T\n",
    "\n",
    "draw_convergence_graph(u, graph_color[convergence_result])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Continuous $u(t)=e^{At}u(0)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "### e.g. A 2×2 system of diff eqs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "$\n",
    "\\begin{cases}\n",
    "\\frac{du_{1}}{dt} = -u_{1}+2u_{2} \\ , \\ u_{1}(0)=1 \\ , \\ u_{1}(t)=? \\\\\n",
    "\\\\\n",
    "\\frac{du_{2}}{dt} = \\ \\ \\ u_{1}-2u_{2} \\ , \\ u_{2}(0)=0 \\ , \\ u_{2}(t)=?\n",
    "\\end{cases}\n",
    "\\\\\n",
    "\\\n",
    "\\\\\n",
    "A = \n",
    "\\begin{bmatrix}\n",
    "-1 & 2 \\\\\n",
    "1 & -2\n",
    "\\end{bmatrix}\n",
    "\\\n",
    ",\n",
    "\\\n",
    "u=\n",
    "\\begin{bmatrix}\n",
    "u_{1} \\\\\n",
    "u_{2}\n",
    "\\end{bmatrix}\n",
    "\\\n",
    "\\rightarrow\n",
    "\\\n",
    "\\frac{du}{dt} = Au\n",
    "\\\n",
    ",\n",
    "\\\n",
    "u(0)=\n",
    "\\begin{bmatrix}\n",
    "1 \\\\\n",
    "0\n",
    "\\end{bmatrix}\n",
    "\\\\\n",
    "\\\n",
    "\\\\\n",
    "u(t)=e^{At}u(0)=Se^{\\Lambda t} c \\ , \\ c = S^{-1} u(0)\n",
    "$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "A = np.array([[-1, 2],\n",
    "              [1, -2]])\n",
    "\n",
    "u_0 = np.array([1, 0], dtype='float')\n",
    "\n",
    "\n",
    "def u(t_):\n",
    "    return expm(A * t_) @ u_0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "$\n",
    "u_{1}(100)=?\n",
    "\\\\\n",
    "u_{2}(100)=?\n",
    "$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0.7]\n",
      " [0.3]]\n"
     ]
    }
   ],
   "source": [
    "print(u(100).reshape(2, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "### e.g. A 2nd order diff eq"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "$\n",
    "y^{''}+by^{'}+ky=0\n",
    "\\\n",
    "\\Rightarrow\n",
    "\\\n",
    "\\begin{cases}\n",
    "y^{''}=-by^{'}-ky\n",
    "\\\\\n",
    "y^{'}= \\ \\ \\ \\ \\ y^{'}+0\n",
    "\\end{cases}\n",
    "\\\n",
    "\\Rightarrow\n",
    "\\\n",
    "\\begin{bmatrix}\n",
    "y^{''}\n",
    "\\\\\n",
    "y^{'}\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "-b & -k \\\\\n",
    "1 & 0\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "y^{'}\n",
    "\\\\\n",
    "y\n",
    "\\end{bmatrix}\n",
    "\\\n",
    "\\Rightarrow\n",
    "\\\n",
    "\\frac{du}{dt}=u^{'}=Au\n",
    "\\\n",
    "\\Rightarrow\n",
    "\\\n",
    "u(t)=e^{At}u(0)\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "### Convergence"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "#### Stability"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "$\\forall i: Re\\lambda_{i}<0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def matrix_exponential_leads_to_stability(la_):\n",
    "    return (la_.real <= -0.1).all()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "#### Steady State"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "$\\exists i: Re\\lambda_{i}=0, \\forall j: Re\\lambda_{j} \\leq 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def matrix_exponential_leads_to_steady_state(la_):\n",
    "    return (la_.real < 0.1).all() and (-0.1 < la_.real).any()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "#### Blow Up"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "$\\exists i: Re\\lambda_{i}>0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def matrix_exponential_blows_up(la_):\n",
    "    return (la_.real >= 0.1).any()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "### Analyze"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A\n",
      "[[1 2 1]\n",
      " [3 8 1]\n",
      " [0 4 1]]\n",
      "\n",
      "u_0\n",
      "[[1.]\n",
      " [1.]\n",
      " [1.]]\n"
     ]
    }
   ],
   "source": [
    "A = create_my_array()\n",
    "\n",
    "u_0 = np.ones(A.shape[0])\n",
    "\n",
    "print('A', A, '', 'u_0', u_0[:, np.newaxis], sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "la\n",
      "[9.4+0.j 0.3+1.j 0.3-1.j]\n"
     ]
    }
   ],
   "source": [
    "la = npLA.eigvals(A)\n",
    "print('la', la, sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001B[91mIt blows up!\u001B[0m\n",
      "\n",
      "u_t\n",
      "[[inf]\n",
      " [inf]\n",
      " [inf]]\n"
     ]
    }
   ],
   "source": [
    "t = 100\n",
    "e_to_the_At = expm(A * t)\n",
    "u_t = e_to_the_At @ u_0\n",
    "\n",
    "convergence_result = ''\n",
    "\n",
    "if matrix_exponential_leads_to_stability(la):\n",
    "    assert_norm_of_last_vector_is_by_far_smaller_than_first_one(u_0, u_t)\n",
    "\n",
    "    convergence_result = 'stability'\n",
    "\n",
    "if matrix_exponential_leads_to_steady_state(la):\n",
    "    assert_norm_of_last_vector_is_not_far_away_than_first_one(u_0, u_t)\n",
    "\n",
    "    convergence_result = 'steady_state'\n",
    "\n",
    "if matrix_exponential_blows_up(la):\n",
    "    assert_norm_of_last_vector_is_by_far_bigger_than_first_one(u_0, u_t)\n",
    "\n",
    "    convergence_result = 'blow_up'\n",
    "\n",
    "print(convergence_message[convergence_result])\n",
    "\n",
    "# print('', 'e_to_the_At', e_to_the_At, sep='\\n')\n",
    "\n",
    "print('', 'u_t', u_t[:, np.newaxis], sep='\\n')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "#### Graph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "<Figure size 504x504 with 1 Axes>",
      "image/png": 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\n"
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "u = [u_0]\n",
    "\n",
    "dt = .1\n",
    "for i in range(int(t / dt)):\n",
    "    du = A @ u[-1] * dt\n",
    "    u.append(u[-1] + du)\n",
    "\n",
    "u = np.array(u).T\n",
    "draw_convergence_graph(u, graph_color[convergence_result])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "[&#x21e7;](#Contents)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "# Markov matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0.1 0.  0.3]\n",
      " [0.2 1.  0.3]\n",
      " [0.7 0.  0.4]]\n"
     ]
    }
   ],
   "source": [
    "A = create_a_Markov_matrix()\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "la\n",
      "[-0.2  0.7  1. ]\n",
      "\n",
      "S\n",
      "[[ 0.7 -0.3 -0. ]\n",
      " [ 0.1  0.8 -1. ]\n",
      " [-0.7 -0.5 -0. ]]\n"
     ]
    }
   ],
   "source": [
    "la, S = npLA.eig(A)\n",
    "print('la', la, '', 'S', S, sep='\\n')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## All entries are 0 or positive"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def assert_all_entries_are_0_or_positive(M):\n",
    "    assert_(\n",
    "        (M >= 0).all()\n",
    "    )\n",
    "\n",
    "\n",
    "assert_all_entries_are_0_or_positive(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## All columns add to 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def assert_all_columns_add_to_1(M):\n",
    "    assert_allclose(\n",
    "        np.sum(M, axis=0).all(),\n",
    "        1\n",
    "    )\n",
    "\n",
    "\n",
    "assert_all_columns_add_to_1(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## $\\exists i: \\lambda_{i} = 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def assert_at_least_an_eigenvalue_is_close_to_1(la_):\n",
    "    assert_(\n",
    "        np.isclose(la_, 1).any()\n",
    "    )\n",
    "\n",
    "\n",
    "assert_at_least_an_eigenvalue_is_close_to_1(la)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## $\\forall j \\ne i , |\\lambda_{j}| \\leq 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "def assert_all_eigenvalues_are_less_than_or_close_to_1(la_):\n",
    "    assert_array_less(\n",
    "        np.absolute(la_),\n",
    "        np.full_like(la_, 1.1),\n",
    "    )\n",
    "\n",
    "\n",
    "assert_all_eigenvalues_are_less_than_or_close_to_1(la)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## It converges to a positive steady state"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "k = 100\n",
    "A_to_the_k = matrix_power(A, k)\n",
    "\n",
    "u_0 = np.ones(A.shape[0])\n",
    "u_k = A_to_the_k @ u_0\n",
    "\n",
    "la = npLA.eigvals(A_to_the_k)\n",
    "\n",
    "assert_array_less(np.zeros_like(u_k), u_k)\n",
    "assert_norm_of_last_vector_is_not_far_away_than_first_one(u_0, u_k)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## $A^{k}$ is still Markov"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "assert_all_entries_are_0_or_positive(A_to_the_k)\n",
    "assert_all_columns_add_to_1(A_to_the_k)\n",
    "assert_at_least_an_eigenvalue_is_close_to_1(la)\n",
    "assert_all_eigenvalues_are_less_than_or_close_to_1(la)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "[&#x21e7;](#Contents)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "# Expansion with a basis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Vector\n",
      "[1 2 3]\n"
     ]
    }
   ],
   "source": [
    "v = np.array([1, 2, 3])\n",
    "print('Vector', v, sep='\\n')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## With A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 1]\n",
      " [3 8 1]\n",
      " [0 4 1]]\n"
     ]
    }
   ],
   "source": [
    "A = create_a_square_invertible_array()\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-1.   0.5  1. ]\n"
     ]
    }
   ],
   "source": [
    "x = inv(A.T @ A) @ A.T @ v\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "assert_allclose(\n",
    "    A @ x,\n",
    "    v\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## With Q (orthonormal basis)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.2  0.1  1. ]\n",
      " [-0.9  0.4 -0.3]\n",
      " [-0.4 -0.9 -0. ]]\n"
     ]
    }
   ],
   "source": [
    "Q = LA.orth(A)\n",
    "print(Q)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Cofactors on Q\n",
      "[-3.2 -1.9  0.4]\n"
     ]
    }
   ],
   "source": [
    "x = Q.T @ v\n",
    "print('Cofactors on Q', x, sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "assert_allclose(\n",
    "    Q @ x,\n",
    "    v\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## With S (eigenvectors)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "la\n",
      "[9.4+0.j 0.3+1.j 0.3-1.j]\n",
      "\n",
      "S\n",
      "[[ 0.3+0.j   0. -0.5j  0. +0.5j]\n",
      " [ 0.9+0.j  -0.1+0.2j -0.1-0.2j]\n",
      " [ 0.4+0.j   0.8+0.j   0.8-0.j ]]\n"
     ]
    }
   ],
   "source": [
    "la, S = npLA.eig(A)\n",
    "print('la', la, '', 'S', S, sep='\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2.8-0.j  1.2+0.2j 1.2-0.2j]\n"
     ]
    }
   ],
   "source": [
    "x = inv(S.T @ S) @ S.T @ v\n",
    "print(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "assert_allclose(\n",
    "    S @ (x * la),\n",
    "    A @ v\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "[&#x21e7;](#Contents)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "# Complex Vectors and Matrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.+0.j 0.+1.j]\n"
     ]
    }
   ],
   "source": [
    "z = np.array([1, 1j])\n",
    "print(z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "length_squared = z.conj().T @ z\n",
    "\n",
    "assert_allclose(\n",
    "    length_squared,\n",
    "    LA.norm(z) ** 2\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Hermitian matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.+0.j  0.+2.j  1.+1.j]\n",
      " [-0.-2.j  8.+0.j  4.-3.j]\n",
      " [ 1.-1.j  4.+3.j -3.+0.j]]\n"
     ]
    }
   ],
   "source": [
    "A = create_a_Hermitian_array()\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "assert_allclose(\n",
    "    A.conj().T,\n",
    "    A\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Unitary matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0.5+0.5j 0.5-0.5j]\n",
      " [0.5-0.5j 0.5+0.5j]]\n"
     ]
    }
   ],
   "source": [
    "Q = create_a_unitary_array()\n",
    "print(Q)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [],
   "source": [
    "assert_allclose(\n",
    "    inv(Q),\n",
    "    Q.conj().T\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "[&#x21e7;](#Contents)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "# Positive Definite Matrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1 2 1]\n",
      " [3 8 1]\n",
      " [0 4 1]]\n"
     ]
    }
   ],
   "source": [
    "A = create_my_array()\n",
    "print(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "la = LA.eigvals(A)\n",
    "(la > 0).all()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 133,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U = LA.lu(A)[2]\n",
    "pivots = np.diag(U)\n",
    "(pivots > 0).all()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {
    "pycharm": {
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": "True"
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sub_dets_of_A = np.array([det(A[:i, :i]) for i in range(1, A.shape[0] + 1)])\n",
    "(sub_dets_of_A > 0).all()"
   ]
  },
  {
   "cell_type": "markdown",
   "source": [
    "[&#x21e7;](#Contents)"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python (LA)",
   "language": "python",
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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