【国科大——矩阵分析与应用】求矩阵的秩，并判断方程是否可解。

矩阵分析与应用——求矩阵的秩，并判断方程是否可解

$$\begin{array}{c}\{\begin{array}{l}0.835x+0.667y=0.168\\ 0.333x+0.266y=0.067\end{array}\end{array}$$
记矩阵 A 和 (A |b) 为此线性方程组的系数矩阵和增广矩阵。分别使用 5 个 和 6 个有效数字计算矩阵 A 和 (A |b) 的 Rank，并判断方程组是否可解。

import numpy as np

def Rank_CoefficientMatrix(Mat, num):
    if (Mat.shape[0] != Mat.shape[1]):
        raise ValueError("size of matrix doesn't match!")
    A = Mat.copy()
    arg = '.' + str(num) + 'g'
    l_ik = float(format(A[1, 0] / A[0, 0], arg))
    # print(l_ik)
    a1 = float(format(A[0, 0], arg))
    b1 = float(format(A[0, 1], arg))
    # print('b1:',b1)
    a2 = float(format(A[1, 0], arg))
    b2 = float(format(A[1, 1], arg))
    b2 = float(format(b2 - float(format(l_ik * b1, arg)), arg))
    if (b2 == 0):
        return 1
    else:
        return 2


def Rank_AugmentedMatrix(Mat, Bais, num):
    if (Mat.shape[0] != Mat.shape[1]) or (Mat.shape[0] != Bais.shape[0]):
        raise ValueError("size of matrix doesn't match!")
    A = Mat.copy()
    B = Bais.copy()
    arg = '.' + str(num) + 'g'
    l_ik = float(format(A[1, 0] / A[0, 0], arg))
    # print(l_ik)
    a1 = float(format(A[0, 0], arg))
    b1 = float(format(A[0, 1], arg))
    a2 = float(format(A[1, 0], arg))
    b2 = float(format(A[1, 1], arg))
    c1 = float(format(B[0, 0], arg))
    c2 = float(format(B[1, 0], arg))
    b2 = float(format(b2 - float(format(l_ik * b1, arg)), arg))
    c2 = float(format(c2 - float(format(l_ik * c1, arg)), arg))
    if(b2==0 and  c2==0):
        return 1
    else:
        return 2

def test(x, y, num):

    print('系数矩阵的秩:', Rank_CoefficientMatrix(x, num))
    print('增广矩阵的秩:', Rank_AugmentedMatrix(x, y, num))
    if (Rank_CoefficientMatrix(x, num) == Rank_AugmentedMatrix(x, y, num) and Rank_AugmentedMatrix(x, y, num) == x.shape[1]):
        print('方程组有唯一解')
    if (Rank_CoefficientMatrix(x, num) == Rank_AugmentedMatrix(x, y, num) and Rank_AugmentedMatrix(x, y, num) < x.shape[1]):
        print('方程组有多个解')
    if (Rank_CoefficientMatrix(x, num) < Rank_AugmentedMatrix(x, y, num)):
        print('方程组无解')

if __name__ == '__main__':
    x = np.mat([[0.835, 0.667],
                [0.333, 0.266]])
    y = np.mat([[0.168],
                [0.067]])
    print(test(x,y,5))
    print('---------')
    print(test(x,y,6))