Editing Unitary transformation

{{Use American English|date=January 2019}}{{Short description|Endomorphism preserving the inner product
}}
{{other uses|Transformation (mathematics) (disambiguation)}}
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 [[isometry|isometric isomorphism]] between two [[inner product space]]s (such as [[Hilbert space]]s). 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 operator==
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 transformation==
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 number|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]]. 
 
==See also==
*[[Antiunitary]]
*[[Orthogonal transformation]]
*[[T-symmetry|Time reversal]]
*[[Unitary group]]
*[[Unitary operator]]
*[[Unitary matrix]]
*[[Wigner's theorem]]
*[[Unitary transformation (quantum mechanics)|Unitary transformations in quantum mechanics]]

[[Category:Linear algebra]]
[[Category:Functional analysis]]