Template:Use American EnglishTemplate:Short description {{#invoke:other uses|otheruses}} 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 definitionEdit
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
- <math>U : H_1 \to H_2</math>
between two inner product spaces, <math>H_1</math> and <math>H_2,</math> such that
- <math>\langle Ux, Uy \rangle_{H_2} = \langle x, y \rangle_{H_1} \quad \text{ for all } x, y \in H_1.</math>
It is a linear isometry, as one can see by setting <math>x=y.</math>
Unitary operatorEdit
In the case when <math>H_1</math> and <math>H_2</math> are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator.
Antiunitary transformationEdit
A closely related notion is that of antiunitary transformation, which is a bijective function
- <math>U:H_1\to H_2\,</math>
between two complex Hilbert spaces such that
- <math>\langle Ux, Uy \rangle = \overline{\langle x, y \rangle}=\langle y, x \rangle</math>
for all <math>x</math> and <math>y</math> in <math>H_1</math>, where the horizontal bar represents the complex conjugate.