unitary matrix in nLab

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Definition

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 applying complex conjugation to all its entries takes it to its inverse:

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

  hence equivalently:

  U⋅U †=I U \cdot U^\dagger \;=\; \mathrm{I}

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

• quaternion unitary group