### 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.

## Contents

- Formal definition 1
- Properties 2
- Unitary operator 3
- Antiunitary transformation 4
- See also 5

## Formal definition

More precisely, a **unitary transformation** is an isomorphism between two Hilbert spaces. In other words, a *unitary transformation* is a bijective function

- U:H_1\to H_2\,

where H_1 and H_2 are Hilbert spaces, such that

- \langle Ux, Uy \rangle_{H_2} = \langle x, y \rangle_{H_1}

for all x and y in H_1.

## Properties

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

## Unitary operator

In the case when H_1 and 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

- U:H_1\to H_2\,

between two complex Hilbert spaces such that

- \langle Ux, Uy \rangle = \overline{\langle x, y \rangle}=\langle y, x \rangle

for all x and y in H_1, where the horizontal bar represents the complex conjugate.

## See also

- Antiunitary
- Orthogonal transformation
- Time reversal
- Unitary group
- Unitary operator
- Unitary matrix
- Wigner's Theorem