nLab
unitary matrix

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Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

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Homotopy groups

Basic facts

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Definition

An n×nn \times n-matrix UMat(n,)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 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:

    UU =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).

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