**Conjugate transpose of a matrix** ‘**P’** is basically a matrix which is equal to the **conjugate** of the matrix obtained by taking the **transpose** of the matrix ‘**P’**. In order to find the conjugate transpose of any matrix; firstly, transpose is obtained and secondly, the conjugate is obtained. The conjugate transpose is generally denoted as $P^{\theta} or P^* or (\overline{P})^T$.

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## Conjugate transpose of a matrix example

Let P is a matrix such that

\[P=\begin{bmatrix} 1+i &2+3i \\ 4-2i& 6 \end{bmatrix}\]

Now, to find the conjugate transpose of this matrix P, we first find the transpose of matrix P i.e.

\[P^T=\begin{bmatrix} 1+i &4-2i \\ 2+3i& 6 \end{bmatrix}\]

In the second step, we find conjugate of the matrix P^{T}

\[P^{\theta}= P^* = (\overline{P})^T=\begin{bmatrix} 1-i &4+2i \\ 2-3i& 6 \end{bmatrix}\]

This is the conjugate transpose of a 2 x 2 matrix P.

## Conjugate transpose of a matrix properties

The conjugate transpose of matrices S and R are S^{θ} and R^{θ}, respectively. Then,

1. (S^{θ})^{θ} = S and (R^{θ})^{θ} = R

2. (S + R)^{θ} = S^{θ} + R^{θ}

$3. (aS)^{\theta}=\overline{a}S^{\theta} and (aR)^{\theta}=\overline{a}R^{\theta}$

where ‘**a**‘ is a complex number.

4. (SR)^{θ} = R^{θ}S^{θ}

The conjugate transpose is used to check special complex matrices i.e. hermitian matrix, skew hermitian matrix and unitary matrix.