unitary transformation

{{Use American English|date=January 2019}}{{Short description|Endomorphism preserving the inner product

}}

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.

Antiunitary transformation

A closely related notion is that of antiunitary transformation, which is a bijective function

: $U:H_1\to H_2\,$

between two complex Hilbert spaces such that

: $\langle Ux, Uy \rangle = \overline{\langle x, y \rangle}=\langle y, x \rangle$

for all $x$ and $y$ in $H_1$ , where the horizontal bar represents the complex conjugate.

unitary transformation

Formal definition

Unitary operator

Antiunitary transformation

See also