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 (such as Hilbert spaces). In other words, a unitary transformation is a bijective function

U:H_{1}\to H_{2}

between two inner product spaces, $H_{1}$ and $H_{2},$ such that

\langle Ux,Uy\rangle _{H_{2}}=\langle x,y\rangle _{H_{1}}\quad {\text{ for all }}x,y\in H_{1}.

It is a linear isometry, as one can see by setting $x=y.$

Unitary operator

In the case when $H_{1}$ and $H_{2}$ are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator.