【线代】矩阵转置性质及代码证明

矩阵转置的定义

定义： 把矩阵A的行换成同序列数的列得到一个新矩阵，叫做A的转置矩阵 ，记作$A^T$

矩阵转置的性质

1.$(A^T)^T=A$

import numpy as np
A = np.random.randint(0, 100, [3, 3])
print((A.T).T == A)
[OUT]:
[[ True  True  True]
 [ True  True  True]
 [ True  True  True]]

2.$(A+B)^T=A^T+B^T$

import numpy as np
A = np.random.randint(0, 100, [3, 3])
B = np.random.randint(0, 100, [3, 3])
print((A+B).T==A.T+B.T)
[OUT]:
[[ True  True  True]
 [ True  True  True]
 [ True  True  True]]

3.$(\lambda A)^T=\lambda A^T$

import numpy as np
A = np.random.randint(0, 100, [3, 3])
lambda_ = 3.14
print((lambda_*A).T == lambda_*A.T)
[OUT]:
[[ True  True  True]
 [ True  True  True]
 [ True  True  True]]

4.$(AB)^T=B^TA^T$

import numpy as np
A = np.random.randint(0, 100, [2, 4])
B = np.random.randint(0, 100, [4, 2])
print((A@B).T == B.T@A.T)
[OUT]:
[[ True  True]
 [ True  True]]

5. 若方阵A满足$A^T=A$，则称A为对称矩阵。$A=(a_{ij})_n$为对称矩阵的充要条件是$a_{ij}=a_{ji}(i, j=1, 2, \cdots, n)$

import numpy as np

def symmetric(shape):
    matrix = np.triu(np.random.randint(0, 100, shape))
    matrix += matrix.T-np.diag(matrix.diagonal())
    return matrix
    
A = symmetric([3, 3])
print(A.T == A)
[OUT]:
[[ True  True  True]
 [ True  True  True]
 [ True  True  True]]