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.

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 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 also[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]