Biorthogonal system

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

Contents

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

Related to a biorthogonal system is the projection

where its image is the linear span of and the kernel is

Construction

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

where is the matrix with entries

See also

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References

  1. Bhushan, Datta, Kanti (2008). Matrix And Linear Algebra, Edition 2: AIDED WITH MATLAB. PHI Learning Pvt. Ltd. p. 239. ISBN   9788120336186.