Orthogonal basis

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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 coordinates[edit]

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 analysis[edit]

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


Symmetric bilinear form[edit]

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 form[edit]

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 also[edit]


  • 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 links[edit]