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 $\mathbf A$ is also denoted by $\mathbf A^*$ or $\mathbf A^{\mathrm H}$.

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

Also see

 * Definition:Adjoint Operator