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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.

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

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



Related to a biorthogonal system is the projection


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


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 alsoEdit


  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]