{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from numpy.linalg import eig\n",
    "\n",
    "%precision 4\n",
    "np.set_printoptions(precision=4, suppress=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Probability matrix.\n",
    "P = np.array([[0.70, 0.15, 0.15],\n",
    "              [0.20, 0.80, 0.15],\n",
    "              [0.10, 0.05, 0.70]])\n",
    "\n",
    "P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "eigenvalues, eigenvectors = eig(P)\n",
    "eigenvalues"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "eigenvectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = np.array([0.15, 0.20, 0.65])\n",
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy.linalg import solve\n",
    "\n",
    "# Compute coefficients a for X as a \n",
    "# linear combination of eigenvectors: X = a@eigenvectors.\n",
    "a = solve(eigenvectors, X)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Verify the linear combination.\n",
    "t = np.zeros(len(a))\n",
    "\n",
    "# Compute vector t = a@eigenvectors\n",
    "for i in range(len(a)):\n",
    "    print(f'{a[i]}*{eigenvectors[:,i]}')\n",
    "    t = t + a[i]*eigenvectors[:, i]\n",
    "\n",
    "print()\n",
    "print(f't = {t}')\n",
    "print(f'X = {X}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "size = len(eigenvalues)\n",
    "t = np.zeros(size)\n",
    "    \n",
    "for i in range(size):\n",
    "    t = t + a[i]*(eigenvalues[i]**22)*eigenvectors[:,i]\n",
    "    \n",
    "t"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Probability matrix.\n",
    "P = np.array([[0.5, 0.3, 0.1, 0.1],\n",
    "              [0.3, 0.4, 0.2, 0.1],\n",
    "              [0.2, 0.3, 0.2, 0.1],\n",
    "              [0.0, 0.0, 0.5, 0.7]])\n",
    "\n",
    "P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Initial state (percentage of employees in each rank).\n",
    "X = np.array([0.30, 0.35, 0.25, 0.10])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import MatrixPowers as mp\n",
    "\n",
    "mp.matrix_power(P, 68, X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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