In mathematics, a unitary transformation is a linear isomorphism that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.

Formal definition

More precisely, a unitary transformation is an isometric isomorphism between two inner product spaces (such as Hilbert spaces). In other words, a unitary transformation is a bijective function

${\displaystyle U:H_{1}\to H_{2))$

between two inner product spaces, ${\displaystyle H_{1))$ and ${\displaystyle H_{2},}$ such that

${\displaystyle \langle Ux,Uy\rangle _{H_{2))=\langle x,y\rangle _{H_{1))\quad {\text{ for all ))x,y\in H_{1}.}$

It is a linear isometry, as one can see by setting ${\displaystyle x=y.}$

Unitary operator

In the case when ${\displaystyle H_{1))$ and ${\displaystyle H_{2))$ are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator.

Antiunitary transformation

A closely related notion is that of antiunitary transformation, which is a bijective function

${\displaystyle U:H_{1}\to H_{2}\,}$

between two complex Hilbert spaces such that

${\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle ))=\langle y,x\rangle }$

for all ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle H_{1))$, where the horizontal bar represents the complex conjugate.