Conjugate Transpose of a Matrix – Example and Properties

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 $P^{\theta} or P^* or (\overline{P})^T$.

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^θS^θ

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

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