nLab unitary matrix

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).

Related concepts

• quaternion unitary group