Subsection 2.3.3 Complex - Conjugate of a Matrix, \(A^{*}\)
The matrix formed by taking the complex - conjugate of each element of any matrix A. It is denoted by \(A^{*}\) or \(\bar{A}\text{.}\) Hence we have \(\bar{A}= \bar{a}_{ij}\) (for all \(i\) and \(j\)). For example: let the matrix,
\begin{equation*}
A= \begin{bmatrix} 1+i & 2-3i & 4 \\ 7+2i & -i & 3-2i \end{bmatrix}
\end{equation*}
then, it’s conjugate matrix is
\begin{equation*}
\bar{A} = \begin{bmatrix}
1-i & 2+3i & 4\\7-2i & i & 3+2i
\end{bmatrix}.
\end{equation*}
If \(\bar{A}=A\text{,}\) then A is a real matrix.
Properties of Complex-Conjugate Matrix.
- The conjugate of the conjugate of matrix \(A\) is the matrix \(A\) i.e., \((A^{*})^{*}=A\text{.}\)
- The complex conjugate of the sum of two matrices is the sum of their complex conjugations, i.e., \((A+B)^{*}= A^{*}+B^{*}\text{.}\)
- The conjugate of scalar multiple of a matrix is the scalar multiple of its comples conjugate, i.e., \((\alpha A^{*})= \alpha^{*} A^{*}\text{,}\) where \(\alpha\) is a complex number. or, \((k A^{*})= k A^{*}\text{,}\) where \(k\) is a real scalar number.
- The conjugate of the product of two matrices is the product of their conjugates in the same order, i.e., \((AB)^{*}=A^{*}B^{*}\text{.}\)
