Conjugate of a Matrix – Example and Properties

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.

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