---
tag: 0EX045E2BP
kind: definition
name: Conjugate transpose
msc: [15A99, 15B57]
defines: 0DXKE10PHF
uses: [0W809K6AW0]
sources:
  - url: https://en.wikipedia.org/wiki/Conjugate_transpose
    locator: Definition
---

## Definition

Let $A = (a_{ij})$ be an $m \times n$ matrix with complex entries.
The *conjugate transpose* of $A$ is the $n \times m$ matrix $A^*$ with entries
$$(A^*)_{ij} = \overline{a_{ji}}$$
for $1 \le i \le n$ and $1 \le j \le m$, where $\overline{z}$ is the complex conjugate of $z$.

## Remarks

Other notations are $A^H$ and $A^\dagger$.
For a real matrix, the conjugate transpose is the [[transpose|09DXWYRD43]].
The rules $(A^*)^* = A$ and $(AB)^* = B^* A^*$ hold.
For the standard inner product on $\C^n$, $\langle Ax, y \rangle = \langle x, A^* y \rangle$.

## Explanations

### L1 human

The conjugate transpose flips a complex matrix over its diagonal and conjugates each entry.
For example, the conjugate transpose of $\begin{pmatrix} 1 & i \\ 0 & 2 \end{pmatrix}$ is $\begin{pmatrix} 1 & 0 \\ -i & 2 \end{pmatrix}$.
For complex matrices it plays the role that the transpose plays for real matrices.