Eigenvalues

computing_eigenvalues.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import var, Matrix\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_eigenvalues(n):\n",
    "    p = var('p', positive=True)\n",
    "    q = 1 - 2*p \n",
    "    row = [q, p]\n",
    "    row.extend(np.zeros(n - 3))\n",
    "    row.append(p)\n",
    "    m = Matrix([list(np.roll(row, i)) for i in range(n)])\n",
    "    return m.eigenvals()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "def print_eigenlist(eigendict):\n",
    "    n = sum(eigendict.values())\n",
    "    print(\"n = {}. \".format(n) + \"=\"*70)\n",
    "    for (k, v) in eigendict.items():\n",
    "        print(\"Multiplicity: {} \\t \\t Eigenvalue: {}\".format(v, k))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "n = 4. ======================================================================\n",
      "Multiplicity: 1 \t \t Eigenvalue: 1\n",
      "Multiplicity: 1 \t \t Eigenvalue: -4*p + 1\n",
      "Multiplicity: 2 \t \t Eigenvalue: -2*p + 1\n",
      "n = 5. ======================================================================\n",
      "Multiplicity: 1 \t \t Eigenvalue: 1\n",
      "Multiplicity: 2 \t \t Eigenvalue: -5*p/2 + sqrt(5)*p/2 + 1\n",
      "Multiplicity: 2 \t \t Eigenvalue: -5*p/2 - sqrt(5)*p/2 + 1\n",
      "n = 6. ======================================================================\n",
      "Multiplicity: 1 \t \t Eigenvalue: 1\n",
      "Multiplicity: 2 \t \t Eigenvalue: -3*p + 1\n",
      "Multiplicity: 1 \t \t Eigenvalue: -4*p + 1\n",
      "Multiplicity: 2 \t \t Eigenvalue: -p + 1\n",
      "n = 7. ======================================================================\n",
      "Multiplicity: 1 \t \t Eigenvalue: 1\n",
      "Multiplicity: 2 \t \t Eigenvalue: -7*p/3 - (-42*p**2 + 42*p + (7*p - 3)**2 - 9)/(3*(-1/2 + sqrt(3)*I/2)*(189*p**3/2 - 189*p**2 + 189*p/2 + (7*p - 3)**3 - (63*p - 27)*(14*p**2 - 14*p + 3)/2 + sqrt(-4*(-42*p**2 + 42*p + (7*p - 3)**2 - 9)**3 + (189*p**3 - 378*p**2 + 189*p + 2*(7*p - 3)**3 - (63*p - 27)*(14*p**2 - 14*p + 3) - 27)**2)/2 - 27/2)**(1/3)) - (-1/2 + sqrt(3)*I/2)*(189*p**3/2 - 189*p**2 + 189*p/2 + (7*p - 3)**3 - (63*p - 27)*(14*p**2 - 14*p + 3)/2 + sqrt(-4*(-42*p**2 + 42*p + (7*p - 3)**2 - 9)**3 + (189*p**3 - 378*p**2 + 189*p + 2*(7*p - 3)**3 - (63*p - 27)*(14*p**2 - 14*p + 3) - 27)**2)/2 - 27/2)**(1/3)/3 + 1\n",
      "Multiplicity: 2 \t \t Eigenvalue: -7*p/3 - (-42*p**2 + 42*p + (7*p - 3)**2 - 9)/(3*(-1/2 - sqrt(3)*I/2)*(189*p**3/2 - 189*p**2 + 189*p/2 + (7*p - 3)**3 - (63*p - 27)*(14*p**2 - 14*p + 3)/2 + sqrt(-4*(-42*p**2 + 42*p + (7*p - 3)**2 - 9)**3 + (189*p**3 - 378*p**2 + 189*p + 2*(7*p - 3)**3 - (63*p - 27)*(14*p**2 - 14*p + 3) - 27)**2)/2 - 27/2)**(1/3)) - (-1/2 - sqrt(3)*I/2)*(189*p**3/2 - 189*p**2 + 189*p/2 + (7*p - 3)**3 - (63*p - 27)*(14*p**2 - 14*p + 3)/2 + sqrt(-4*(-42*p**2 + 42*p + (7*p - 3)**2 - 9)**3 + (189*p**3 - 378*p**2 + 189*p + 2*(7*p - 3)**3 - (63*p - 27)*(14*p**2 - 14*p + 3) - 27)**2)/2 - 27/2)**(1/3)/3 + 1\n",
      "Multiplicity: 2 \t \t Eigenvalue: -7*p/3 - (-42*p**2 + 42*p + (7*p - 3)**2 - 9)/(3*(189*p**3/2 - 189*p**2 + 189*p/2 + (7*p - 3)**3 - (63*p - 27)*(14*p**2 - 14*p + 3)/2 + sqrt(-4*(-42*p**2 + 42*p + (7*p - 3)**2 - 9)**3 + (189*p**3 - 378*p**2 + 189*p + 2*(7*p - 3)**3 - (63*p - 27)*(14*p**2 - 14*p + 3) - 27)**2)/2 - 27/2)**(1/3)) - (189*p**3/2 - 189*p**2 + 189*p/2 + (7*p - 3)**3 - (63*p - 27)*(14*p**2 - 14*p + 3)/2 + sqrt(-4*(-42*p**2 + 42*p + (7*p - 3)**2 - 9)**3 + (189*p**3 - 378*p**2 + 189*p + 2*(7*p - 3)**3 - (63*p - 27)*(14*p**2 - 14*p + 3) - 27)**2)/2 - 27/2)**(1/3)/3 + 1\n",
      "n = 8. ======================================================================\n",
      "Multiplicity: 1 \t \t Eigenvalue: 1\n",
      "Multiplicity: 1 \t \t Eigenvalue: -4*p + 1\n",
      "Multiplicity: 2 \t \t Eigenvalue: -2*p - sqrt(2)*p + 1\n",
      "Multiplicity: 2 \t \t Eigenvalue: -2*p + 1\n",
      "Multiplicity: 2 \t \t Eigenvalue: -2*p + sqrt(2)*p + 1\n",
      "n = 9. ======================================================================\n",
      "Multiplicity: 1 \t \t Eigenvalue: 1\n",
      "Multiplicity: 2 \t \t Eigenvalue: -2*p - (-27*p**2 + 36*p + (6*p - 3)**2 - 9)/(3*(-1/2 + sqrt(3)*I/2)*(81*p**3/2 - 243*p**2/2 + 81*p + (6*p - 3)**3 - (54*p - 27)*(9*p**2 - 12*p + 3)/2 + sqrt(-4*(-27*p**2 + 36*p + (6*p - 3)**2 - 9)**3 + (81*p**3 - 243*p**2 + 162*p + 2*(6*p - 3)**3 - (54*p - 27)*(9*p**2 - 12*p + 3) - 27)**2)/2 - 27/2)**(1/3)) - (-1/2 + sqrt(3)*I/2)*(81*p**3/2 - 243*p**2/2 + 81*p + (6*p - 3)**3 - (54*p - 27)*(9*p**2 - 12*p + 3)/2 + sqrt(-4*(-27*p**2 + 36*p + (6*p - 3)**2 - 9)**3 + (81*p**3 - 243*p**2 + 162*p + 2*(6*p - 3)**3 - (54*p - 27)*(9*p**2 - 12*p + 3) - 27)**2)/2 - 27/2)**(1/3)/3 + 1\n",
      "Multiplicity: 2 \t \t Eigenvalue: -3*p + 1\n",
      "Multiplicity: 2 \t \t Eigenvalue: -2*p - (-27*p**2 + 36*p + (6*p - 3)**2 - 9)/(3*(81*p**3/2 - 243*p**2/2 + 81*p + (6*p - 3)**3 - (54*p - 27)*(9*p**2 - 12*p + 3)/2 + sqrt(-4*(-27*p**2 + 36*p + (6*p - 3)**2 - 9)**3 + (81*p**3 - 243*p**2 + 162*p + 2*(6*p - 3)**3 - (54*p - 27)*(9*p**2 - 12*p + 3) - 27)**2)/2 - 27/2)**(1/3)) - (81*p**3/2 - 243*p**2/2 + 81*p + (6*p - 3)**3 - (54*p - 27)*(9*p**2 - 12*p + 3)/2 + sqrt(-4*(-27*p**2 + 36*p + (6*p - 3)**2 - 9)**3 + (81*p**3 - 243*p**2 + 162*p + 2*(6*p - 3)**3 - (54*p - 27)*(9*p**2 - 12*p + 3) - 27)**2)/2 - 27/2)**(1/3)/3 + 1\n",
      "Multiplicity: 2 \t \t Eigenvalue: -2*p - (-27*p**2 + 36*p + (6*p - 3)**2 - 9)/(3*(-1/2 - sqrt(3)*I/2)*(81*p**3/2 - 243*p**2/2 + 81*p + (6*p - 3)**3 - (54*p - 27)*(9*p**2 - 12*p + 3)/2 + sqrt(-4*(-27*p**2 + 36*p + (6*p - 3)**2 - 9)**3 + (81*p**3 - 243*p**2 + 162*p + 2*(6*p - 3)**3 - (54*p - 27)*(9*p**2 - 12*p + 3) - 27)**2)/2 - 27/2)**(1/3)) - (-1/2 - sqrt(3)*I/2)*(81*p**3/2 - 243*p**2/2 + 81*p + (6*p - 3)**3 - (54*p - 27)*(9*p**2 - 12*p + 3)/2 + sqrt(-4*(-27*p**2 + 36*p + (6*p - 3)**2 - 9)**3 + (81*p**3 - 243*p**2 + 162*p + 2*(6*p - 3)**3 - (54*p - 27)*(9*p**2 - 12*p + 3) - 27)**2)/2 - 27/2)**(1/3)/3 + 1\n"
     ]
    }
   ],
   "source": [
    "for i in range(4, 10):\n",
    "    print_eigenlist(get_eigenvalues(i))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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