Editing Isometry (section)

=== Linear isometry ===

Given two [[normed vector space]]s <math> V </math> and <math> W ,</math> a '''linear isometry''' is a [[linear map]] <math> A : V \to W </math> that preserves the norms:
:<math>\|Av\|_W = \|v\|_V </math>
for all <math>v \in V.</math><ref name="Thomsen 2017 p125">{{cite book |last=Thomsen |first=Jesper Funch |year=2017 |title=Lineær algebra |trans-title=Linear Algebra |page=125 |lang=da |location=Århus |publisher=Aarhus University |series=Department of Mathematics}}</ref> 
Linear isometries are distance-preserving maps in the above sense. 
They are global isometries if and only if they are [[surjective]].

In an [[inner product space]], the above definition reduces to 

:<math>\langle v, v \rangle_V = \langle Av, Av \rangle_W </math>

for all <math>v \in V,</math> which is equivalent to saying that <math>A^\dagger A = \operatorname{Id}_V.</math> This also implies that isometries preserve inner products, as

:<math>\langle A u, A v \rangle_W = \langle u, A^\dagger A v \rangle_V = \langle u, v \rangle_V</math>.

Linear isometries are not always [[unitary operator]]s, though, as those require additionally that <math> V = W </math> and <math> A A^\dagger = \operatorname{Id}_V</math> (i.e. the [[Domain_of_a_function|domain]] and [[codomain]] coincide and <math> A </math> defines a [[Unitary operator|coisometry]]).

By the [[Mazur–Ulam theorem]], any isometry of normed vector spaces over <math> \mathbb{R} </math> is [[Affine transformation|affine]].

A linear isometry also necessarily preserves angles, therefore a linear isometry transformation is a [[conformal linear transformation]].

;Examples

* A [[linear map]] from <math> \mathbb{C}^n </math> to itself is an isometry (for the [[dot product]]) if and only if its matrix is [[unitary matrix|unitary]].<ref>
{{cite journal
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 | doi = 10.1126/science.290.5500.2323
 | journal = [[Science (journal)|Science]]
 | volume = 290 | issue = 5500 | pages = 2323–2326
 | pmid =  11125150 | bibcode = 2000Sci...290.2323R
 | citeseerx = 10.1.1.111.3313
}}
</ref><ref>
{{cite journal
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 |last2=Roweis |first2=Sam T.
 |date=June 2003
 | title= Think globally, fit locally: Unsupervised learning of nonlinear manifolds
 |journal=[[Journal of Machine Learning Research]]
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 |quote=Quadratic optimisation of <math>\mathbf{M}=(I-W)^\top(I-W)</math> (page 135) such that <math>\mathbf{M}\equiv YY^\top</math>
}}
</ref><ref name=Zhang-Zha-2004/><ref name=Zhang-Wang-2006/>