Illustrating matrix transpose rules in matrix multiplication

| categories: linear algebra | View Comments

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;