Orthogonal basis

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

As coordinates

Any orthogonal basis can be used to define a system of orthogonal coordinates Template:Mvar. 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

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

Extensions

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

$\langle \mathbf {e} _{j},\mathbf {e} _{k}\rangle =\left\{{\begin{array}{ll}q(\mathbf {e} _{k})&j=k\\0&j\neq k\end{array}}\right.\quad ,$

where Template:Mvar is a quadratic form associated with Template:Langle·,·Template:Rangle: q(v) = Template:Langlev, vTemplate:Rangle (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 Template:Mvar and Template:Mvar are components of v and w in {e_k} .

References

• Template:Lang Algebra
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External links

• Weisstein, Eric W., "Orthogonal Basis", MathWorld.

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