Editing Unitary matrix (section)

{{Short description|Complex matrix whose conjugate transpose equals its inverse}}
{{for multi|matrices with  orthogonality  over the real number field|orthogonal matrix|the restriction on the allowed evolution of quantum systems that ensures the sum of probabilities of all possible outcomes of any event always equals 1|unitarity}}

In [[linear algebra]], an [[invertible matrix|invertible]] [[Complex number|complex]] [[Matrix (mathematics)|square matrix]] {{mvar|U}} is '''unitary''' if its [[Invertible matrix|matrix inverse]] {{math|''U''<sup>−1</sup>}} equals its [[conjugate transpose]] {{math|''U''<sup>*</sup>}}, that is, if

<math display=block>U^* U = UU^* = I,</math>

where {{mvar|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 (mark)|dagger]] ({{tmath|\dagger}}), so the equation above is written

<math display=block>U^\dagger U = UU^\dagger = I.</math>

A complex matrix {{mvar|U}} is '''special unitary''' if it is unitary and its [[matrix determinant]] equals {{math|1}}.

For [[real number]]s, the analogue of a unitary matrix is an [[orthogonal matrix]]. Unitary matrices have significant importance in quantum mechanics because they preserve [[Norm (mathematics)|norms]], and thus, [[probability amplitude]]s.