The transpose of the matrix \(A \in \mathbb{R}^{m \times n}\) is \(A^T \in \mathbb{R}\) where
\begin{equation*} (A)_{i,j} = (A^T)_{j,i}\text{.} \end{equation*}However, if \(A \in \mathbb{C}^{m \times n}\) we need to define the related concept of the Hermetian Transpose, \(A^\star\text{,}\) for which
\begin{equation*} (A)_{i,j} = \overline{(A^T)_{j,i}}\text{.} \end{equation*}Here \(\bar{z}\) denotes the complex conjugate of \(z\text{.}\) It is easy to show that
\begin{equation*} (AB)^\star = B^\star A^\star. \end{equation*} in-context