Hermitian Transposition

Definition: Hermitian Transposition

The Hermitian transpose or conjugate transpose $A^{†}$ of a complex matrix $A\in {\mathbb{C}}^{m\times n}$ is the transpose of $A$ combined with the complex conjugation of every entry.

$${\left[\begin{array}{ccc}{a}_{11}& \cdots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{m1}& \cdots & a_{mn}\end{array}\right]}^{†}\stackrel{\text{def}}{=}\left[\begin{array}{ccc}{\bar{a}}_{11}& \cdots & {\bar{a}}_{m1}\\ ⋮& \ddots & ⋮\\ {\bar{a}}_{1n}& \cdots & {\bar{a}}_{mn}\end{array}\right]$$

NOTATION

The Hermitian transpose of $A$ is usually denoted by $A^{†}$ or $A^{\ast }$.