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.
More precisely, a unitary transformation is an isomorphism between two Hilbert spaces. In other words, a unitary transformation is a bijective function
where and are Hilbert spaces, such that
for all and in .
A unitary transformation is an isometry, as one can see by setting in this formula.
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.
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.