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.

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

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 ⁠${\displaystyle \left\{e_{k}\right\}}$⁠: $${\displaystyle \langle e_{j},e_{k}\rangle ={\begin{cases}q(e_{k})&j=k\\0&j\neq k,\end{cases}}}$$ where ${\displaystyle q}$ is a quadratic form associated with ${\displaystyle \langle \cdot ,\cdot \rangle :}$ ${\displaystyle q(v)=\langle v,v\rangle }$ (in an inner product space, ⁠${\displaystyle q(v)=\Vert v\Vert ^{2}}$⁠).

Hence for an orthogonal basis ⁠${\displaystyle \left\{e_{k}\right\}}$⁠, $${\displaystyle \langle v,w\rangle =\sum _{k}q(e_{k})v_{k}w_{k},}$$ where ${\displaystyle v_{k}}$ and ${\displaystyle w_{k}}$ are components of ${\displaystyle v}$ and ${\displaystyle w}$ in the basis.

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 ⁠${\displaystyle q(v+w)-q(v)-q(w)=0}$⁠.

• 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

• 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.

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