---
tag: 0R9R0Y1HDK
kind: object
name: Unitary group
msc: [15B57, 20G20]
instance_of: [G5S1M9]
uses: [0D6RMJ10M9]
sources:
  - url: https://en.wikipedia.org/wiki/Unitary_group
    locator: lead section
same_as:
  wikidata: Q1500617
  wikipedia: Unitary group
---

## Description

Let $n \ge 1$.
The *unitary group* is the set
$$U(n) = \{U \in M_n(\C) : U^* U = I_n\}$$
of complex $n \times n$ [[unitary matrices|0AR68JTMDT]], with matrix multiplication.
It is a subgroup of the general linear group $GL_n(\C)$.

## Remarks

It is the group of linear isometries of $\C^n$ with the standard inner product.
$U(1)$ is the circle group $\{z \in \C : \abs{z} = 1\}$.
The unitary matrices with determinant $1$ form the special unitary group $SU(n)$.
$U(n)$ is a compact connected Lie group of real dimension $n^2$.

## Explanations

### L1 human

The unitary group collects all complex matrices that keep the length of every vector.
For $n = 1$, these are multiplications by complex numbers of absolute value $1$, that is, rotations of the complex plane.
The group appears throughout physics.