In mathematics, a biorthogonal system is a pair of indexed families of vectors such that where and 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 and is an orthonormal system.

Projection

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Related to a biorthogonal system is the projection   where   its image is the linear span of   and the kernel is  

Construction

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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

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References

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  1. ^ Bhushan, Datta, Kanti (2008). Matrix And Linear Algebra, Edition 2: AIDED WITH MATLAB. PHI Learning Pvt. Ltd. p. 239. ISBN 9788120336186.{{cite book}}: CS1 maint: multiple names: authors list (link)
  • Jean Dieudonné, On biorthogonal systems Michigan Math. J. 2 (1953), no. 1, 7–20 [1]