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, ${\displaystyle \mathbf {A} ^{\mathrm {H} }=\mathbf {A} ^{\operatorname {T} }}$.

The conjugate transpose of an ${\displaystyle m\times n}$ matrix ${\displaystyle \mathbf {A} }$ is formally defined by

where the subscript ${\displaystyle ij}$ denotes the ${\displaystyle (i,j)}$-th entry (matrix element), for ${\displaystyle 1\leq i\leq n}$ and ${\displaystyle 1\leq j\leq m}$, and the overbar denotes a scalar complex conjugate.

This definition can also be written as

where ${\displaystyle \mathbf {A} ^{\operatorname {T} }}$ denotes the transpose and ${\displaystyle {\overline {\mathbf {A} }}}$ denotes the matrix with complex conjugated entries.

Other names for the conjugate transpose of a matrix are Hermitian transpose, Hermitian conjugate, adjoint matrix or transjugate. The conjugate transpose of a matrix ${\displaystyle \mathbf {A} }$ can be denoted by any of these symbols:

In some contexts, ${\displaystyle \mathbf {A} ^{*}}$ denotes the matrix with only complex conjugated entries and no transposition.