Orthogonal basis
# Orthogonal basis

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

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

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

## Extensions

### Symmetric bilinear form

The concept of an orthogonal basis is applicable to a vector space $V$ (over any field) equipped with a symmetric bilinear form $\langle \cdot ,\cdot \rangle ,$ where orthogonality of two vectors $v$ and $w$ means $\langle v,w\rangle =0.$ For an orthogonal basis $\left\{e_{k}\right\}:$

$\langle e_{j},e_{k}\rangle ={\begin{cases}q(e_{k})&j=k\\0&j\neq k,\end{cases}}$

where $q$ is a quadratic form associated with $\langle \cdot ,\cdot \rangle :$ $q(v)=\langle v,v\rangle$ (in an inner product space, $q(v)=\|v\|^{2}.$).

Hence for an orthogonal basis $\left\{e_{k}\right\},$

$\langle v,w\rangle =\sum _{k}q(e_{k})v^{k}w^{k},$

where $v_{k}$ and $w_{k}$ are components of $v$ and $w$ in the basis.

### Quadratic form

The concept of orthogonality may be extended to a vector space (over any field) equipped with a quadratic form $q(v)$. Starting from the observation that, when the characteristic of the underlying field is not 2, the associated symmetric bilinear form $\langle v,w\rangle ={\tfrac {1}{2}}(q(v+w)-q(v)-q(w))$ allows vectors $v$ and $w$ to be defined as being orthogonal with respect to $q$ when $q(v+w)-q(v)-q(w)=0.$

