unitary matrix (Rev #3, changes) in nLab

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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 applyingcomplex conjugation to all its entries takes it to itself: its inverse:

  U †=UU −1. U^\dagger = U U^{-1} \,.

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