---
tag: 0D6RMJ10M9
kind: definition
name: Unitary matrix
msc: [15B57, 15A21]
defines: 0AR68JTMDT
uses: [0EX045E2BP, 06DW9BAWB1]
sources:
  - url: https://en.wikipedia.org/wiki/Unitary_matrix
    locator: lead section
---

## Definition

Let $n \ge 1$ and let $U$ be a complex $n \times n$ matrix.
The matrix $U$ is *unitary* when
$$U^* U = I_n,$$
where $U^*$ is the [[conjugate transpose|0DXKE10PHF]] and $I_n$ is the [[identity matrix|06DW9BAWB1]].

## Remarks

Equivalently, $U$ is invertible with $U^{-1} = U^*$, and then also $U U^* = I_n$.
Equivalently, the columns of $U$ form an orthonormal basis of $\C^n$ for the standard inner product.
The eigenvalues of a unitary matrix have absolute value $1$, and $\abs{\det U} = 1$.
A real unitary matrix is an [[orthogonal matrix|0W02QESPJQ]].

## Explanations

### L1 human

Unitary matrices are the complex version of orthogonal matrices.
They keep all lengths and inner products in $\C^n$.
In quantum mechanics, they describe how a closed system evolves in time.