Unitary matrix

In linear algebra, an invertiblecomplexsquare matrixU is unitary if its matrix inverseU equals its conjugate transposeU, that is, if

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

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 (⁠${\displaystyle \dagger }$⁠), so the equation above is written

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

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 the normalization of state vectors and the inner products between them.