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.
More precisely, a unitary transformation is an isometric isomorphism between two inner product spaces (such as Hilbert spaces). In other words, a unitary transformation is a bijective function
between two inner product spaces, and such that
It is a linear isometry, as one can see by setting
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.