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 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 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.

Also see

• Definition:Adjoint Operator

• Results about Hermitian conjugates can be found here.

Source of Name

This entry was named for Charles Hermite.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): adjoint: 1. a.
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Hermitian conjugate
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hermitian conjugate
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Hermitian conjugate (Hermitian adjoint)