Definition:Hermitian Conjugate

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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 defined and denoted:

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

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

That is, $\mathbf A^\dagger$ is the transpose of the complex conjugate of $\mathbf A$.

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

Also see

  • Results about Hermitian conjugates can be found here.

Source of Name

This entry was named for Charles Hermite.