Orthogonal basis

Summary

In mathematics, particularly linear algebra, an orthogonal basis for an inner product space is a basis for whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis.

As coordinatesEdit

Any orthogonal basis can be used to define a system of orthogonal coordinates   Orthogonal (not necessarily orthonormal) bases are important due to their appearance from curvilinear orthogonal coordinates in Euclidean spaces, as well as in Riemannian and pseudo-Riemannian manifolds.

In functional analysisEdit

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

ExtensionsEdit

Symmetric bilinear formEdit

The concept of an orthogonal basis is applicable to a vector space   (over any field) equipped with a symmetric bilinear form   where orthogonality of two vectors   and   means   For an orthogonal basis  

 
where   is a quadratic form associated with     (in an inner product space,  ).

Hence for an orthogonal basis  

 
where   and   are components of   and   in the basis.

Quadratic formEdit

The concept of orthogonality may be extended to a vector space (over any field) equipped with a quadratic form  . Starting from the observation that, when the characteristic of the underlying field is not 2, the associated symmetric bilinear form   allows vectors   and   to be defined as being orthogonal with respect to   when  

See alsoEdit

ReferencesEdit

  • Lang, Serge (2004), Algebra, Graduate Texts in Mathematics, vol. 211 (Corrected fourth printing, revised third ed.), New York: Springer-Verlag, pp. 572–585, ISBN 978-0-387-95385-4
  • Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. p. 6. ISBN 3-540-06009-X. Zbl 0292.10016.

External linksEdit