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 natural ...
s). In other words, a ''unitary transformation'' is a
bijective function
between two inner product spaces,
and
such that
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'' ...
, as one can see by setting
in this formula.
Unitary operator
In the case when
and
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
:
between two
complex Hilbert spaces such that
:
for all
and
in
, where the horizontal bar represents the
complex conjugate.
See also
*
Antiunitary
*
Orthogonal transformation
*
Time reversal
*
Unitary group
*
Unitary operator
*
Unitary matrix
*
Wigner's theorem
*
Unitary transformations in quantum mechanics
Linear algebra
Functional analysis
ru:Унитарное преобразование