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 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 $\mathbf A$ 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 Hermitian transpose
  • the conjugate transpose
  • the associate matrix
  • the adjoint matrix, in the context of quantum mechanics
  • the Hermitian adjoint

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

Source of Name

This entry was named for Charles Hermite.

Also see