Orthogonal basis


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

Any orthogonal basis can be used to define a system of orthogonal coordinates. Orthogonal 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 using multiplication by nonzero scalars.

Extensions

The concept of an orthogonal basis is applicable to a vector space equipped with a symmetric bilinear form, where orthogonality of two vectors and means. For an orthogonal basis :
where is a quadratic form associated with : .
Hence for an orthogonal basis,
where and are components of and in the basis.