# Unitary matrix

## Complex matrix whose conjugate transpose equals its inverse / From Wikipedia, the free encyclopedia

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In linear algebra, an invertible complex square matrix U is **unitary** if its matrix inverse *U*^{−1} equals its conjugate transpose *U*^{*}, that is, if

$U^{*}U=UU^{*}=I,$

where I is the identity matrix.

In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (†), so the equation above is written

$U^{\dagger }U=UU^{\dagger }=I.$

A complex matrix U is **special unitary** if it is unitary and its matrix determinant equals 1.

For real numbers, the analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.