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 Hermitian transpose, conjugate transpose or adjoint matrix.

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.