Matrix Transpose

This page introduces transposes of real and complex matrices.

Real Matrices

The transpose of an $m\times n$ matrix of real numbers

is given by

That means, when taking the transpose of $A$,

• the $i$-th row of $A$ becomes the $i$-th column of $A'$
• the $j$-th column of $A$ becomes the $j$-th row of $A'$

for all $i=1,2,\dots,m$ and all $j=1,2,\ldots,n$,

To find transposes, SIMO offers the following operators and functions:

• .' (transpose)
• transpose() (transpose)
• ' (complex conjugate transpose)
• ctranspose() (complex conjugate transpose)

The complex conjugate of a real number is itself. Therefore, for the real matrix A, the following matrices are the same:

• A'
• A.'
• transpose(A)
• ctranspose(A)

💡 These operators and functions do not work for A with more than 2 dimensions, i.e., ndim(A) > 2.

Example

The transpose and conjugate transpose of a $3\times 4$ randomly generated real matrix are the same.

Input

% Random matrix.
r = rand(3,4)
% Transpose.
r.'
% Complex conjugate transpose
r'

Output

 = 1e-1 × 
 0.7546   8.7160   3.0373   5.5844
 3.9817   0.0541   4.3378   5.1050
 8.1191   7.9236   3.2574   7.7646

ans = 1e-1 × 
 0.7546   3.9817   8.1191
 8.7160   0.0541   7.9236
 3.0373   4.3378   3.2574
 5.5844   5.1050   7.7646

ans = 1e-1 × 
 0.7546   3.9817   8.1191
 8.7160   0.0541   7.9236
 3.0373   4.3378   3.2574
 5.5844   5.1050   7.7646

Complex Matrices

The conjugate transpose of an $m\times n$ matrix of complex numbers

is given by

For the matrix A of complex numbers, its conjugate transpose can be obtained by A' or ctranspose(A). When conjugates are not required, you can use A.' or transpose(A) to find the (non-conjugate) transpose.

💡 These operators and functions do not work for A with more than 2 dimensions, i.e., ndim(A) > 2.

Example

The transpose and complex conjugate transpose of a $3\times 4$ randomly generated complex matrix.

Input

% Random complex matrix.
r = rand(3,4) + rand(3,4)*i
% Transpose.
r.'
% Complex conjugate transpose
r'

Output

r = 1e-1 × 
 6.4637 + 6.6888i   2.5425 + 8.0223i   7.3450 + 8.2596i   2.6829 + 7.0021i
 4.0511 + 0.3917i   7.4915 + 5.3168i   8.9667 + 0.0910i   4.4199 + 6.4485i
 4.6943 + 5.5860i   0.8644 + 5.4784i   1.0989 + 8.8801i   8.5181 + 2.9303i

ans = 1e-1 × 
 6.4637 + 6.6888i   4.0511 + 0.3917i   4.6943 + 5.5860i
 2.5425 + 8.0223i   7.4915 + 5.3168i   0.8644 + 5.4784i
 7.3450 + 8.2596i   8.9667 + 0.0910i   1.0989 + 8.8801i
 2.6829 + 7.0021i   4.4199 + 6.4485i   8.5181 + 2.9303i

ans = 1e-1 × 
 6.4637 - 6.6888i   4.0511 - 0.3917i   4.6943 - 5.5860i
 2.5425 - 8.0223i   7.4915 - 5.3168i   0.8644 - 5.4784i
 7.3450 - 8.2596i   8.9667 - 0.0910i   1.0989 - 8.8801i
 2.6829 - 7.0021i   4.4199 - 6.4485i   8.5181 - 2.9303i