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 $\alpha_{j i}$.

The Hermitian conjugate of a matrix $\mathbf A$ can also be seen denoted by:

$\mathbf A^*$

$\mathbf A'$

$\mathbf A^{\mathrm H}$

$\mathbf A^{\bot}$

The Hermitian conjugate is also known as:

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

This entry was named for Charles Hermite.