Documentation

# ctranspose, '

Symbolic matrix complex conjugate transpose

## Syntax

``A'``
``ctranspose(A)``

## Description

example

````A'` computes the complex conjugate transpose of `A`.```
````ctranspose(A)` is equivalent to `A'`.```

## Examples

### Conjugate 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 complex conjugate 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 to its nonconjugate transform.

`isAlways(A' == A.')`
```ans = 3×2 logical array 1 1 1 1 1 1```

### Conjugate 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 conjugate transpose of this matrix. The complex conjugate transpose operator, `A'`, performs a transpose and negates the sign of the imaginary portion of the complex elements in `A`.

`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 complex conjugate transform is not equal to the nonconjugate transform.

`isAlways(A' == A.','Unknown','false')`
```ans = 2×2 logical array 0 0 0 0```

## Input Arguments

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Input, specified as a number or a symbolic number, variable, expression, vector, matrix, multidimensional array.

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### Complex Conjugate Transpose

The complex conjugate transpose of a matrix interchanges the row and column index for each element, reflecting the elements across the main diagonal. The operation also negates the imaginary part of any complex numbers.

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