Unitary matrix

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, an <a href="/facts/Invertible_matrix/fPqXk3V8">invertible</a> <a href="/facts/Complex_number/2w7mqI7e">complex</a> <a href="/facts/Matrix_(mathematics)/qa8FI4ko">square matrix</a> U is unitary if its <a href="/facts/Invertible_matrix/fPqXk3V8">matrix inverse</a> U−1 equals its <a href="/facts/Conjugate_transpose/7eBWtqVG">conjugate transpose</a> U*, that is, if

 
 
 
 
 U
 
 ∗
 
 
 U
 =
 U
 
 U
 
 ∗
 
 
 =
 I
 ,
 
 
 {\displaystyle U^{*}U=UU^{*}=I,}

where I is the <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>.
In <a href="/facts/Physics/a7aH2y08">physics</a>, especially in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, the conjugate transpose is referred to as the <a href="/facts/Hermitian_adjoint/gN6RHphd">Hermitian adjoint</a> of a matrix and is denoted by a <a href="/facts/Dagger_(mark)/lDNNgbgr">dagger</a> (⁠
 
 
 
 †
 
 
 {\displaystyle \dagger }
 
⁠), so the equation above is written

 
 
 
 
 U
 
 †
 
 
 U
 =
 U
 
 U
 
 †
 
 
 =
 I
 .
 
 
 {\displaystyle U^{\dagger }U=UU^{\dagger }=I.}

A complex matrix U is special unitary if it is unitary and its <a href="/facts/Matrix_determinant/eGYqJ2TF">matrix determinant</a> equals 1.
For <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, the analogue of a unitary matrix is an <a href="/facts/Orthogonal_matrix/WDXBXYB5">orthogonal matrix</a>. Unitary matrices have significant importance in quantum mechanics because they preserve <a href="/facts/Norm_(mathematics)/xIbR4uE1">norms</a>, and thus, <a href="/facts/Probability_amplitude/ryPgJvs8">probability amplitudes</a>.

Unitary matrix open-in-new

Unitary matrix