Biorthogonal system

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In mathematics, a biorthogonal system is a pair of indexed families of vectors

in E and in F

such that

where E and F form a pair of topological vector spaces that are in duality, ⟨,⟩ is a bilinear mapping and is the Kronecker delta.

A biorthogonal system in which E = F and is an orthonormal system.

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

Projection[edit]

Related to a biorthogonal system is the projection

,

where ; its image is the linear span of , and the kernel is .

Construction[edit]

Given a possibly non-orthogonal set of vectors and the projection related is

,

where is the matrix with entries .

  • , and then is a biorthogonal system.

See also[edit]

References[edit]

  1. ^ Bhushan, Datta, Kanti (2008). Matrix And Linear Algebra, Edition 2: AIDED WITH MATLAB. PHI Learning Pvt. Ltd. p. 239. ISBN 9788120336186. 
  • Jean Dieudonné, On biorthogonal systems Michigan Math. J. 2 (1953), no. 1, 7–20 [1]