# Definition:Hermitian Conjugate

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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 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**, **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.

## Source of Name

This entry was named for Charles Hermite.

## Also see

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**adjoint**:**1. a.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Hermitian conjugate**