# Unitary transformation

In mathematics, a unitary transformation is a transformation 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 isomorphism between two Hilbert spaces. In other words, a unitary transformation is a bijective function

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

where ${\displaystyle H_}$ and ${\displaystyle H_}$ are Hilbert spaces, such that

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

for all ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle H_}$.

## Properties

A unitary transformation is an isometry, as one can see by setting ${\displaystyle x=y}$ in this formula.

## Unitary operator

In the case when ${\displaystyle H_}$ and ${\displaystyle H_}$ 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

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

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 ${\displaystyle H_}$, where the horizontal bar represents the complex conjugate.