Contents

# Contents

## Definition

An $n \times n$-matrix $U \in Mat(n, \mathbb{C})$ with entries in the complex numbers (for $n$ a natural number) is unitary if the following equivalent conditions hold

• it preserves the canonical inner product on $\mathbb{C}^n$;

• the operation $(-)^\dagger$ of transposing it and then applying complex conjugation to all its entries takes it to its inverse:

$U^\dagger \;=\; U^{-1} \,.$

hence equivalently:

$U \cdot U^\dagger \;=\; \mathrm{I}$

For fixed $n$, the unitary matrices under matrix product form a Lie group: the unitary group $\mathrm{U}(n)$ (or other notations).

Last revised on August 13, 2020 at 05:34:37. See the history of this page for a list of all contributions to it.