Conjugate of a matrix is the matrix obtained from matrix ‘P’ on replacing its elements with the corresponding conjugate complex numbers. It is denoted by $\overline{P}$.
Conjugate of a matrix example
Let Q is a matrix such that
\[Q=\begin{bmatrix} 1+i &2+3i \\ 4-2i& 6 \end{bmatrix}\]
Now, to find the conjugate of this matrix Q, we find the conjugate of each element of matrix Q i.e.
\[\overline{Q}=\begin{bmatrix} 1-i &2-3i \\ 4+2i& 6 \end{bmatrix}\]
This is the conjugate of a 2 x 2 matrix Q.
Conjugate of a matrix properties
The conjugate of matrices P and Q are
$\overline{P} \; and \; \overline{Q}, respectively$. Then,
$1. \; \overline{(\overline{P})}=P \; and \; \overline{(\overline{Q})}=Q$
$2. \; \overline{(P+Q)}=\overline{P}+\overline{Q}$
$3. \; \overline{(PQ)}=\overline{P} \; \overline{Q}$
$4. \; \overline{P}=-P$
If P is a purely imaginary matrix
$5. \; \overline{P}=P$
If P is a real matrix
$6. \; \overline{(aP)}=\overline{a} \; \overline{P}$
where ‘a’ is any complex number.
