transpose, .'

Symbolic matrix transpose

Syntax

A.'

transpose(A)

Description

A.' computes the nonconjugate transpose of A.

transpose(A) is equivalent to A.'.

Examples

Transpose of Real Matrix

Create a 2-by-3 matrix, the elements of which represent real numbers.

syms x y real
A = [x x x; y y y]

A =
[ x, x, x]
[ y, y, y]

Find the nonconjugate transpose of this matrix.

A.'

ans =
[ x, y]
[ x, y]
[ x, y]

If all elements of a matrix represent real numbers, then its complex conjugate transform equals its nonconjugate transform.

isAlways(A' == A.')

ans =
  3×2 logical array
     1     1
     1     1
     1     1

Transpose of Complex Matrix

Create a 2-by-2 matrix, the elements of which represent complex numbers.

syms x y real
A = [x + y*i x - y*i; y + x*i y - x*i]

A =
[ x + y*1i, x - y*1i]
[ y + x*1i, y - x*1i] 

Find the nonconjugate transpose of this matrix. The nonconjugate transpose operator, A.', performs a transpose without conjugation. That is, it does not change the sign of the imaginary parts of the elements.

A.'

ans =
[ x + y*1i, y + x*1i]
[ x - y*1i, y - x*1i]

For a matrix of complex numbers with nonzero imaginary parts, the nonconjugate transform is not equal to the complex conjugate transform.

isAlways(A.' == A','Unknown','false')

ans =
  2×2 logical array
     0     0
     0     0

Input Arguments

A — Input

number | symbolic number | symbolic scalar variable | symbolic matrix variable | symbolic function | symbolic expression | symbolic vector | symbolic matrix | symbolic array

Input, specified as a number, or a symbolic number, scalar variable, matrix variable (since R2021a), function, expression, or vector, matrix, or array of symbolic scalar variables.

More About

Nonconjugate Transpose

The nonconjugate transpose of a matrix interchanges the row and column index for each element, reflecting the elements across the main diagonal. The diagonal elements themselves remain unchanged. This operation does not affect the sign of the imaginary parts of complex elements.

For example, if B = A.' and A(3,2) is 1+1i, then the element B(2,3) is 1+1i.