Conjugate of a Matrix – Example and Properties

by

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}

Contents show
.

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.

facebookShare on Facebook
TwitterTweet
FollowFollow us
Related:
  1. Conjugate Transpose of a Matrix – Example and Properties
  2. What is Zero Matrix or Null Matrix? Examples and Properties
  3. What is Unit Matrix or Identity Matrix? Examples and Properties
  4. What is Symmetric Matrix? Eigenvalues, Properties and Examples
  5. What is Skew Symmetric Matrix? Properties and Examples
  6. What is Hermitian Matrix? Example
  7. What is Diagonal Matrix? Inverse, Examples and Properties
  8. What is Skew Hermitian Matrix? Example
  9. What is Unitary Matrix? Example
  10. Transpose of a Matrix – Properties

Leave a Comment

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Adblocker detected! Please consider reading this notice.

We've detected that you are using AdBlock Plus or some other adblocking software which is preventing the page from fully loading.

We don't have any banner, Flash, animation, obnoxious sound, or popup ad. We do not implement these annoying types of ads!

We need fund to operate the site, and almost all of it comes from our online advertising.

Please add electricalvoice.com to your ad blocking whitelist or disable your adblocking software.

×