【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$A^T$AT

矩阵转置的性质

1. $(A^T)^T=A$(AT)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. $(A+B)^T=A^T+B^T$(A+B)T=AT+BT

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. $(\lambda A)^T=\lambda A^T$(λA)T=λAT

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

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

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