Subsection MCC Matrices and Complex Conjugation

As we did with vectors (Definition CCCV), we can define what it means to take the conjugate of a matrix.

Definition CCM Complex Conjugate of a Matrix

Suppose $A$ is an $m\times n$ matrix. Then the conjugate of $A$, written $\conjugate{A}$ is an $m\times n$ matrix defined by \begin{equation*} \matrixentry{\conjugate{A}}{ij}=\conjugate{\matrixentry{A}{ij}} \end{equation*}

Example CCM Complex conjugate of a matrix

The interplay between the conjugate of a matrix and the two operations on matrices is what you might expect.

Theorem CRMA Conjugation Respects Matrix Addition

Suppose that $A$ and $B$ are $m\times n$ matrices. Then $\conjugate{A+B}=\conjugate{A}+\conjugate{B}$.

Proof

Theorem CRMSM Conjugation Respects Matrix Scalar Multiplication

Suppose that $\alpha\in\complex{\null}$ and $A$ is an $m\times n$ matrix. Then $\conjugate{\alpha A}=\conjugate{\alpha}\conjugate{A}$.

Proof

Theorem CCM Conjugate of the Conjugate of a Matrix

Suppose that $A$ is an $m\times n$ matrix. Then $\conjugate{\left(\conjugate{A}\right)}=A$.

Proof

Finally, we will need the following result about matrix conjugation and transposes later.

Theorem MCT Matrix Conjugation and Transposes

Suppose that $A$ is an $m\times n$ matrix. Then $\conjugate{\left(\transpose{A}\right)}=\transpose{\left(\conjugate{A}\right)}$.

Proof