Orthogonal basis

In mathematics, particularly linear algebra, an orthogonal basis for an inner product space ${\displaystyle V}$ is a basis for ${\displaystyle V}$ whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis.

As coordinates[edit]

Any orthogonal basis can be used to define a system of orthogonal coordinates ${\displaystyle V.}$ 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[edit]

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

Extensions[edit]

Symmetric bilinear form[edit]

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

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

${\displaystyle q}$

${\displaystyle \langle \cdot ,\cdot \rangle :}$

${\displaystyle q(v)=\langle v,v\rangle }$

${\displaystyle q(v)=\Vert v\Vert ^{2}}$

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

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

${\displaystyle v_{k}}$

${\displaystyle w_{k}}$

${\displaystyle v}$

${\displaystyle w}$

Quadratic form[edit]

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

See also[edit]

• Basis (linear algebra) – Set of vectors used to define coordinates
• Orthonormal basis – Specific linear basis (mathematics)
• Orthonormal frame – Euclidean space without distance and angles
• Schauder basis – Computational tool
• Total set – subset of a topological vector space whose linear span is densePages displaying wikidata descriptions as a fallback

References[edit]

• Lang, Serge (2004), Algebra, Graduate Texts in Mathematics, vol. 211 (Corrected fourth printing, revised third ed.), New York: Springer-Verlag, pp. 572–585, ISBN 978-0-387-95385-4
• Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. p. 6. ISBN 3-540-06009-X. Zbl 0292.10016.

External links[edit]

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