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.

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, $q(v) = \Vert v \Vert^2$).

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 of characteristic not 2 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 $q(v+w) - q(v) - q(w) = 0$.