如何用NumPy/LinAlg区分线性方程组无解与无穷多解

当系数矩阵行列式为0时，要区分方程组是无解还是有无穷多解，核心是比较系数矩阵的秩和增广矩阵的秩：

• 若两者秩相等，方程组存在无穷多解
• 若系数矩阵秩小于增广矩阵秩，方程组无解

原代码 #

import numpy as np
from sympy import *
from scipy.linalg import solve


a = float(input("Enter coefficient a: "))
b = float(input("Enter coefficient b: "))
c = float(input("Enter coefficient c: "))
d = float(input("Enter coefficient d: "))
e = float(input("Enter coefficient e: "))
f = float(input("Enter coefficient f: "))
g = float(input("Enter coefficient g: "))
h = float(input("Enter coefficient h: "))
i = float(input("Enter coefficient i: "))
j = float(input("Enter coefficient j: "))
k = float(input("Enter coefficient k: "))
l = float(input("Enter coefficient l: "))

A = np.array([[a, b, c],
              [e, f, g],
              [i, j, k]])
b = np.array([d, h, l])


if (np.linalg.det(A)) != 0:
    print(np.linalg.solve(A, b))
elif (np.linalg.det(A)) == 0:
        print("Infinite or no solutions")

修改后的代码 #

利用np.linalg.matrix_rank计算秩，同时处理浮点数精度问题，避免误判：

import numpy as np

# 输入12个系数
a = float(input("Enter coefficient a: "))
b = float(input("Enter coefficient b: "))
c = float(input("Enter coefficient c: "))
d = float(input("Enter coefficient d: "))
e = float(input("Enter coefficient e: "))
f = float(input("Enter coefficient f: "))
g = float(input("Enter coefficient g: "))
h = float(input("Enter coefficient h: "))
i = float(input("Enter coefficient i: "))
j = float(input("Enter coefficient j: "))
k = float(input("Enter coefficient k: "))
l = float(input("Enter coefficient l: "))

A = np.array([[a, b, c],
              [e, f, g],
              [i, j, k]])
b_vec = np.array([d, h, l])

# 处理浮点数精度，用绝对值判断行列式是否接近0
det_A = np.linalg.det(A)
tolerance = 1e-9

if abs(det_A) > tolerance:
    print("方程组有唯一解：", np.linalg.solve(A, b_vec))
else:
    # 构造增广矩阵
    augmented_matrix = np.column_stack((A, b_vec))
    # 计算系数矩阵与增广矩阵的秩
    rank_A = np.linalg.matrix_rank(A, tol=tolerance)
    rank_augmented = np.linalg.matrix_rank(augmented_matrix, tol=tolerance)
    
    if rank_A == rank_augmented:
        print("方程组有无穷多解")
    else:
        print("方程组无解")

关键说明 #

1. 浮点数精度处理：直接判断det(A) == 0会因浮点数计算误差出错，改用绝对值与小容差（如1e-9）比较
2. 秩的计算：np.linalg.matrix_rank支持指定容差，确保秩的计算不受微小数值误差干扰
3. 增广矩阵构造：用np.column_stack将系数矩阵和常数项列合并，得到完整的增广矩阵

内容的提问来源于stack exchange，提问作者Thrylin