det_laplace.py

#!/usr/bin/env python
# -*- coding: utf-8 -*-

from __future__ import print_function
import numpy


# Teorema de Laplace
# Pegando sempre a primeira linha, i = 0
# Cofator: (-1)^(i+j) * MC_ij
# (-1)^(i+j) => 1 se i+j for par e -1 se for ímpar, mas aqui i+j=j
# Menor complementar MC_ij aqui será o determinante da matriz menor formada
# pelas linhas sem a primeira e sem a coluna em análise, assim, da matriz
# transposta sem a primeira linha, "mt", tira-se a linha em análise,
# transpõe novamente e calcula o determinante, multiplicando-o pelo elemento
# correspondente da primeira linha, chegando dessa forma a cada elemento de
# Laplace, e então soma-se tudo para chegar ao determinante por Laplace.
def d(m):
    if len(m) == 1:
        return m[0][0]
    mt = zip(*m[1:])
    return sum(map(lambda (j, e): e * (1,-1)[j%2] * d(zip(*(mt[:j]+mt[j+1:]))),
               enumerate(m[0])))


m2 = [[ 1,  2],
      [-5, -3]]

print("d(m2) =", d(m2), "|", numpy.linalg.det(m2))

m3 = [[ 5, 0,  1],
      [-2, 3,  4],
      [ 0, 2, -1]]

print("d(m3) =", d(m3), "|", numpy.linalg.det(m3))

m4 = [[ 5, 0,  1, 1],
      [-2, 3,  4, 2],
      [ 0, 2, -1, 3],
      [ 1, 2,  3, 4]]

print("d(m4) =", d(m4), "|", numpy.linalg.det(m4))

m5 = [[ 5, 0,  1, 1, 10],
      [-2, 3,  4, 2, 13],
      [ 0, 2, -1, 3, 25],
      [ 1, 2,  3, 4, 18],
      [ 5, 3,  6, 4, -8]]

print("d(m5) =", d(m5), "|", numpy.linalg.det(m5))

m10 = [[ 1.00, -0.50,  0.00,  0.00, -0.50,  0.00,  0.00,  0.00,  0.00,  0.00],
       [-0.33,  1.00, -0.33,  0.00,  0.00, -0.33,  0.00,  0.00,  0.00,  0.00],
       [ 0.00, -0.33,  1.00, -0.33,  0.00,  0.00, -0.33,  0.00,  0.00,  0.00],
       [ 0.00,  0.00, -0.33,  1.00,  0.00,  0.00,  0.00, -0.33,  0.00,  0.00],
       [-0.33,  0.00,  0.00,  0.00,  1.00, -0.33,  0.00,  0.00,  0.00,  0.00],
       [ 0.00, -0.25,  0.00,  0.00, -0.25,  1.00, -0.25,  0.00,  0.00,  0.00],
       [ 0.00,  0.00, -0.25,  0.00,  0.00, -0.25,  1.00, -0.25,  0.00,  0.00],
       [ 0.00,  0.00,  0.00, -0.25,  0.00,  0.00, -0.25,  1.00, -0.25,  0.00],
       [ 0.00,  0.00,  0.00,  0.00,  0.00,  0.00,  0.00, -0.25,  1.00, -0.25],
       [ 0.00,  0.00,  0.00,  0.00,  0.00,  0.00,  0.00,  0.00, -0.25,  1.00]]

print("d(m10) =", d(m10), "|", numpy.linalg.det(m10))