文章目录
- 1 矩阵的逆
- 1.1 求解矩阵的逆
- 2 初等矩阵
- 2.1 初等矩阵和可逆性
- 3 矩阵的LU分解
- 3.1 LU分解的实现
1 矩阵的逆
1.1 求解矩阵的逆
def inv(A):
if A.row_num() != A.col_num():
return None
n = A.row_num()
"""矩阵A+单位矩阵"""
ls = LinearSystem(A, Matrix.identity(n))
"""对线性系统进行高斯消元，如果没有解，返回none"""
if not ls.gauss_jordan_elimination():
return None
"""高斯消元有解的话，把线性系统的右部分取出，重新构成矩阵，得到矩阵的逆"""
invA = [[row[i] for i in range(n, 2*n)] for row in ls.Ab]
return Matrix(invA)
2 初等矩阵
2.1 初等矩阵和可逆性
3 矩阵的LU分解
3.1 LU分解的实现
from .Matrix import Matrix
from .Vector import Vector
from ._globals import is_zero
def lu(matrix):
assert matrix.row_num() == matrix.col_num(), "matrix must be a square matrix"
n = matrix.row_num()
"""A是原矩阵的副本"""
A = [matrix.row_vector(i) for i in range(n)]
"""初始化L，使对角线元素为1"""
L = [[1.0 if i == j else 0.0 for i in range(n)] for j in range(n)]
for i in range(n):
"""看A[i][i]位置是否可以是主元"""
if is_zero(A[i][i]):
return None, None
else: """将主元以下的j位置变为0"""
for j in range(i + 1, n):
p = A[j][i] / A[i][i] """求加减的系数"""
A[j] = A[j] - p * A[i] """将第j行的位置经过加减运算变成0"""
L[j][i] = p """将L矩阵相应位置变成相应变换的值"""
return Matrix(L), Matrix([A[i].underlying_list() for i in range(n)])
