Biorthogonal system explained

In mathematics, a biorthogonal system is a pair of indexed families of vectors$$\backslash tilde\; v\_i\; \backslash text\; E\; \backslash text\; \backslash tilde\; u\_i\; \backslash text\; F$$such that$$\backslash left\backslash langle\backslash tilde\; v\_i,\; \backslash tilde\; u\_j\backslash right\backslash rangle\; =\; \backslash delta\_,$$where

E and F form a pair of topological vector spaces that are in duality, \langle ⋅ , ⋅ \rangle is a bilinear mapping and \delta _{i,j} is the Kronecker delta.

An example is the pair of sets of respectively left and right eigenvectors of a matrix, indexed by eigenvalue, if the eigenvalues are distinct.^[1]

A biorthogonal system in which

E = F and \tilde v _{i} = \tilde u _{i} is an orthonormal system.

Projection

Related to a biorthogonal system is the projection $$P\; :=\; \backslash sum\_\; \backslash tilde\; u\_i\; \backslash otimes\; \backslash tilde\; v\_i,$$where

(u ⊗ v) (x) : = u \langle v, x \rangle; its image is the linear span of \left\{\tilde u _{i:} i \in I\right\}, and the kernel is \left\{\left\langle \tilde v _{i,} ⋅ \right\rangle = 0 : i \in I\right\}.

Construction

Given a possibly non-orthogonal set of vectors

u = \left(u _{i\right)} and v = \left(v _{i\right)} the projection related is $$P\; =\; \backslash sum\_\; u\_i\; \backslash left(\backslash langle\backslash mathbf,\; \backslash mathbf\backslash rangle^\backslash right)\_\; \backslash otimes\; v\_j,$$where \langle v , u \rangle is the matrix with entries \left(\langle v , u \rangle\right) _{i,j} = \left\langle v _{i,} u _{j\right\rangle.} \tilde u _{i} : = (I - P) u _{i,} and \tilde v _{i} : = (I - P) ^{*} v _{i} then is a biorthogonal system.

References

• Jean Dieudonné, On biorthogonal systems Michigan Math. J. 2 (1953), no. 1, 7–20 http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.mmj/1028989861

Notes and References

1. Book: Bhushan. Datta, Kanti. Matrix And Linear Algebra, Edition 2: AIDED WITH MATLAB. 2008. PHI Learning Pvt. Ltd.. 9788120336186. 239. en.