Orthogonal basis

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In mathematics, particularly linear algebra, an orthogonal basis for an inner product space V is a basis for V 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 V. 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.

Extensions[edit]

The concept of an orthogonal (but not of an orthonormal) basis is applicable to a vector space V (over any field) equipped with a symmetric bilinear form ·,·, where orthogonality of two vectors v and w means v, w = 0. For an orthogonal basis {ek} :

\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,

where q is a quadratic form associated with ·,·: q(v) = v, v (in an inner product space q(v) = | v |2). Hence,

\langle\mathbf{v},\mathbf{w}\rangle =
\sum\limits_{k} q(\mathbf{e}_k) v^k w^k\ ,

where vk and wk are components of v and w in {ek} .

References[edit]

External links[edit]