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
More precisely, a unitary transformation is an isomorphism between two Hilbert spaces. In other words, a unitary transformation is a bijective function
where
for all
Properties
A unitary transformation is an isometry, as one can see by setting
Unitary operator
In the case when
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