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.

Copyright (C) 2013 by John Kitchin. See the License for information about copying.

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