Definition:Hermitian Conjugate
(Redirected from Definition:Conjugate Transpose)
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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
- 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)