## Illustrating matrix transpose rules in matrix multiplication

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Illustrating matrix transpose rules in matrix multiplication

# Illustrating matrix transpose rules in matrix multiplication

John Kitchin

## Rules for transposition

Here are the four rules for matrix multiplication and transposition

1. 2. 3. 4. reference: Chapter 7.2 in Advanced Engineering Mathematics, 9th edition. by E. Kreyszig.

## The transpose in Matlab

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

A = [[5 -8 1];
[4 0 0]]

A =

5    -8     1
4     0     0



function

transpose(A)

ans =

5     4
-8     0
1     0



notation

A.'

% note, these functions only provide the non-conjugate transpose. If your
% matrices are complex, then you want the ctranspose function, or the
% notation A' (no dot before the apostrophe). For real matrices there is no
% difference between them.

% below we illustrate each rule using the different ways to get the
% transpose.

ans =

5     4
-8     0
1     0



## Rule 1

m1 = (A.').'
A

all(all(m1 == A)) % if this equals 1, then the two matrices are equal

m1 =

5    -8     1
4     0     0

A =

5    -8     1
4     0     0

ans =

1



## Rule 2

B = [[3 4 5];
[1 2 3]];
m1 = transpose(A+B)
m2 = transpose(A) + transpose(B)

all(all(m1 == m2)) % if this equals 1, then the two matrices are equal

m1 =

8     5
-4     2
6     3

m2 =

8     5
-4     2
6     3

ans =

1



## Rule 3

c = 2.1;
m1 = transpose(c*A)
m2 = c*transpose(A)

all(all(m1 == m2)) % if this equals 1, then the two matrices are equal

m1 =

10.5000    8.4000
-16.8000         0
2.1000         0

m2 =

10.5000    8.4000
-16.8000         0
2.1000         0

ans =

1



## Rule 4

B = [[0 2];
[1 2];
[6 7]]

m1 = (A*B).'
m2 = B.'*A.'

all(all(m1 == m2)) % if this equals 1, then the two matrices are equal

B =

0     2
1     2
6     7

m1 =

-2     0
1     8

m2 =

-2     0
1     8

ans =

1


m3 = A.'*B.'
% you can see m3 has a different shape than m1, so there is no way they can
% be equal.

m3 =

8    13    58
0    -8   -48
0     1     6


% categories: Linear algebra
% tags: math
% post_id = 552; %delete this line to force new post;

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