Definition:Hermitian Conjugate

Definition
Let $\mathbf A = \sqbrk \alpha_{m n}$ be an $m \times n$ matrix over the complex numbers $\C$.

Then the Hermitian conjugate of $\mathbf A$ is denoted $\mathbf A^\dagger$ and is defined as:


 * $\mathbf A^\dagger = \sqbrk \beta_{n m}: \forall i \in \set {1, 2, \ldots, n}, j \in \set {1, 2, \ldots, m}: \beta_{i j} = \overline {\alpha_{j i} }$

where $\overline {\alpha_{j i} }$ denotes the complex conjugate of $\alpha_{j i}$.

Also denoted as
The Hermitian conjugate of a matrix $\mathbf A$ can also be seen denoted by:
 * $\mathbf A^*$
 * $\mathbf A'$
 * $\mathbf A^{\mathrm H}$
 * $\mathbf A^{\bot}$

Also known as
The Hermitian conjugate is also known as the Hermitian transpose, conjugate transpose or adjoint matrix.

The term adjoint matrix is also used for the Hermitian conjugate, so to avoid ambiguity it is recommended that it not be used.

Also see

 * Definition:Adjoint Operator