Unitary transformation
In mathematics, a unitary transformation is a linear isomorphism 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 isometric isomorphism between two inner product spaces. In other words, a unitary transformation is a bijective functionbetween two inner product spaces, and such that
It is a linear isometry, as one can see by setting
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 functionbetween two complex Hilbert spaces such that
for all and in, where the horizontal bar represents the complex conjugate.