\begin{definition}[Definition:Hermitian Conjugate]
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 $\alpha_{j i}$.
\end{definition}