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In [[mathematics]], particularly [[linear algebra]], an '''orthogonal basis''' for an [[inner product space]] {{mvar|V}} is a [[basis (linear algebra)|basis]] for {{mvar|V}} whose vectors are mutually [[orthogonal]].  If the vectors of an orthogonal basis are [[normalize (linear algebra)|normalized]], the resulting basis is an [[orthonormal basis]].


In [[functional analysis]], an orthogonal basis is any basis obtained from an orthonormal basis (or Hilbert basis) using multiplication by nonzero [[Scalar (mathematics)|scalars]].


Any orthogonal basis can be used to define a system of [[orthogonal coordinates]] {{mvar|V}}. Orthogonal (not necessarily orthonormal) bases are important due to their appearance from [[curvilinear coordinates|curvilinear]] orthogonal coordinates in [[Euclidean space]]s, as well as in [[Riemannian manifold|Riemannian]] and [[pseudo-Riemannian manifold|pseudo-Riemannian]] manifolds.
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The concept of an orthogonal (but not of an orthonormal) basis is applicable to a [[vector space]] {{mvar|V}} (over any [[field (mathematics)|field]]) equipped with a [[symmetric bilinear form]] {{math|{{langle}}·,·{{rangle}}}}, where ''[[orthogonality]]'' of two vectors {{math|'''v'''}} and {{math|'''w'''}} means {{math|1={{langle}}'''v''', '''w'''{{rangle}} = 0}}. For an orthogonal basis {{math|{'''e'''<sub>''k''</sub>} }}:
:<math>\langle\mathbf{e}_j,\mathbf{e}_k\rangle =
\left\{\begin{array}{ll}q(\mathbf{e}_k) & j = k \\ 0 & j \ne k
\end{array}\right.\quad,</math>
where {{mvar|q}} is a [[quadratic form]] associated with {{math|{{langle}}·,·{{rangle}}}}: {{math|1=''q''('''v''') = {{langle}}'''v''', '''v'''{{rangle}}}} (in an inner product space {{math|1=''q''('''v''') = {{!}} '''v''' {{!}}<sup>2</sup>}}). Hence,
:<math>\langle\mathbf{v},\mathbf{w}\rangle =
\sum\limits_{k} q(\mathbf{e}_k) v^k w^k\ ,</math>
where {{mvar|v<sup>k</sup>}} and {{mvar|w<sup>k</sup>}} are components of {{math|'''v'''}} and {{math|'''w'''}} in {{math|{'''e'''<sub>''k''</sub>} }}.
 
==References==
{{reflist}}
* {{Lang Algebra | edition=3r2004 | pages=572–585 }}
* {{cite book | first1=J. | last1=Milnor | author1-link=John Milnor| first2=D. | last2=Husemoller | title=Symmetric Bilinear Forms | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]] | volume=73 | publisher=[[Springer-Verlag]] | year=1973 | isbn=3-540-06009-X | zbl=0292.10016 | page=6}}
 
==External links==
* {{MathWorld|title=Orthogonal Basis|urlname=OrthogonalBasis}}
 
[[Category:Linear algebra]]
 
 
{{Linear-algebra-stub}}
{{mathanalysis-stub}}
 
[[de:Orthogonalbasis]]

Latest revision as of 21:30, 30 October 2014


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