In mathematics, a unitary transformation is a transformation that preserves the

inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...

: 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 In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

between two

inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...

s (such as

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...

s). In other words, a ''unitary transformation'' is a bijective function

U : H \to H_2\,

between two inner product spaces,

H

and

H_2,

such that

\langle Ux, Uy \rangle_ = \langle x, y \rangle_ \quad \text x, y \in H.

Properties

A unitary transformation is an

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

, 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

H_1

, where the horizontal bar represents the

complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...

Formal definition

Properties

Unitary operator

Antiunitary transformation

See also