unitary matrix (Rev #2) in nLab

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

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

• the operation (−) †(-)^\dagger of transposing it and then appying complex conjugation to all its entries takes it to itself:

  U †=U. U^\dagger = U \,.

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