Hermitian Transpose#
Definition: Hermitian Transposition
The Hermitian transpose or conjugate transpose \(A^\dagger\) of a complex matrix \(A \in \mathbb{C}^{m \times n}\) is the transpose of \(A\) combined with the complex conjugation of every entry.
\[\begin{bmatrix}a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn}\end{bmatrix}^\dagger \overset{\text{def}}{=} \begin{bmatrix}\bar{a}_{11} & \cdots & \bar{a}_{m1} \\ \vdots & \ddots & \vdots \\ \bar{a}_{1n} & \cdots & \bar{a}_{mn}\end{bmatrix}\]
Notation
The Hermitian transpose of \(A\) is usually denoted by \(A^\dagger\) or \(A^\ast\).