Conjugate transpose

In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an ${\displaystyle m\times n}$ complex matrix ${\displaystyle \mathbf {A} }$ is an ${\displaystyle n\times m}$ matrix obtained by transposing ${\displaystyle \mathbf {A} }$ and applying complex conjugation to each entry (the complex conjugate of ${\displaystyle a+ib}$ being ${\displaystyle a-ib}$, for real numbers ${\displaystyle a}$ and ${\displaystyle b}$). There are several notations, such as ${\displaystyle \mathbf {A} ^{\mathrm {H} }}$ or ${\displaystyle \mathbf {A} ^{*}}$, ${\displaystyle \mathbf {A} '}$, or (often in physics) ${\displaystyle \mathbf {A} ^{\dagger }}$.

For real matrices, the conjugate transpose is just the transpose, ${\displaystyle \mathbf {A} ^{\mathrm {H} }=\mathbf {A} ^{\operatorname {T} }}$.