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.

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.