Subsection 2.3.4 Hermitian Conjugate of a Matrix, \(A^{\dagger}\)
The transpose of the conjugate of a matrix \(A\) is called Hermitian conjugate. It is denoted by \(A^{\dagger}\) or \(A^{\Theta}\text{,}\) e.g.
\begin{equation*}
A= \begin{bmatrix}
1+i & 2-3i & 4 \\ 7+2i & -i & 3-2i \end{bmatrix},
\end{equation*}
\begin{equation*}
A^{*} = \begin{bmatrix} 1-i & 2+3i & 4 \\ 7-2i & i & 3+2i \end{bmatrix},
\end{equation*}
\begin{equation*}
\text{then}\quad A^{\dagger} = (A^{*})^{t}=\begin{bmatrix}
1-i & 7-2i \\ 2+3i & i \\ 4 & 3+2i \end{bmatrix}
\end{equation*}
Properties of Hermitian Conjugate Matrix.
- Transpose conjugate of a matrix is the same as conjugate of its transpose, i.e., \(A^{\dagger} = (A^{*})^{t} = (A^{t})^{*}\text{.}\)
- Transpose conjugate of the conjugate transpose of a matrix is the matrix itself, i.e., \((A^{\dagger})^{\dagger}= A\text{.}\)
- Hermitian conjugate of the sum of two matrices is the sum of their Hermitian conjugate, i.e., \((A+B)^{\dagger} = A^{\dagger} +B^{\dagger}\text{.}\)
- Hermitian conjugate of the product of two matrices is the product of their Hermitian conjugates in reverse order, i.e., \((AB)^{\dagger} = B^{\dagger} A^{\dagger}\text{.}\)
- Hermitian conjugate of scalar multiple of a matrix is the scalar multiple of its comples conjugate, i.e., \((\alpha A)^{\dagger} = \alpha^{*} A^{\dagger}\text{,}\) where \(\alpha\) is a complex number.
