Orthogonal basis explained

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

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

(over any field) equipped with a symmetric bilinear form, where orthogonality of two vectors

and

means . For an orthogonal basis :

\langle e_j, e_k\rangle =\beginq(e_k) & j = k \\0 & j \neq k,\end

where

is a quadratic form associated with

\langle ⋅ , ⋅ \rangle:

q(v)=\langlev,v\rangle

(in an inner product space,).

Hence for an orthogonal basis, $\langle v, w \rangle = \sum_k q(e_k) v^k w^k,$ where

v_k

and

w_k

are components of

and

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 . Starting from the observation that, when the characteristic of the underlying field is not 2, the associated symmetric bilinear form

\langlev,w\rangle=\tfrac{1}{2}(q(v+w)-q(v)-q(w))

allows vectors

and

to be defined as being orthogonal with respect to

when .

References

- Book: J. . Milnor . John Milnor. D. . Husemoller . Symmetric Bilinear Forms . . 73 . . 1973 . 3-540-06009-X . 0292.10016 . 6.