Rules for transposition

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Here are the four rules for matrix multiplication and transposition

1. $$(\mathbf{A}^T)^T = \mathbf{A}$$
2. $$(\mathbf{A}+\mathbf{B})^T = \mathbf{A}^T+\mathbf{B}^T$$
3. $$(\mathit{c}\mathbf{A})^T = \mathit{c}\mathbf{A}^T$$
4. $$(\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T$$

reference: Chapter 7.2 in Advanced Engineering Mathematics, 9th edition. by E. Kreyszig.

1 The transpose in Python

There are two ways to get the transpose of a matrix: with a notation, and with a function.

import numpy as np
A = np.array([[5, -8, 1],
[4, 0, 0]])

# function
print np.transpose(A)

# notation
print A.T

[[ 5  4]
[-8  0]
[ 1  0]]
[[ 5  4]
[-8  0]
[ 1  0]]


2 Rule 1

import numpy as np

A = np.array([[5, -8, 1],
[4, 0, 0]])

print np.all(A == (A.T).T)

True


3 Rule 2

import numpy as np
A = np.array([[5, -8, 1],
[4, 0, 0]])

B = np.array([[3, 4, 5], [1, 2,3]])

print np.all( A.T + B.T == (A + B).T)

True


4 Rule 3

import numpy as np
A = np.array([[5, -8, 1],
[4, 0, 0]])

c = 2.1

print np.all( (c*A).T == c*A.T)

True


5 Rule 4

import numpy as np
A = np.array([[5, -8, 1],
[4, 0, 0]])

B = np.array([[0, 2],
[1, 2],
[6, 7]])

print np.all(np.dot(A, B).T == np.dot(B.T, A.T))

True


6 Summary

That wraps up showing numerically the transpose rules work for these examples.