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Biorthogonal system

Summary

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

{\tilde {v}}_{i}{\text{ in }}E{\text{ and }}{\tilde {u}}_{i}{\text{ in }}F

such that

\left\langle {\tilde {v}}_{i},{\tilde {u}}_{j}\right\rangle =\delta _{i,j},

where

E

and

F

form a pair of topological vector spaces that are in duality,

\langle \,\cdot ,\cdot \,\rangle

is a bilinear mapping and

\delta _{i,j}

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 ${\tilde {v}}_{i}={\tilde {u}}_{i}$ is an orthonormal system.

Projection edit

Related to a biorthogonal system is the projection

P:=\sum _{i\in I}{\tilde {u}}_{i}\otimes {\tilde {v}}_{i},

where

(u\otimes v)(x):=u\langle v,x\rangle ;

its image is the linear span of

\left\{{\tilde {u}}_{i}:i\in I\right\},

and the kernel is

\left\{\left\langle {\tilde {v}}_{i},\cdot \right\rangle =0:i\in I\right\}.

Construction edit

Given a possibly non-orthogonal set of vectors $\mathbf {u} =\left(u_{i}\right)$ and $\mathbf {v} =\left(v_{i}\right)$ the projection related is

P=\sum _{i,j}u_{i}\left(\langle \mathbf {v} ,\mathbf {u} \rangle ^{-1}\right)_{j,i}\otimes v_{j},

where

\langle \mathbf {v} ,\mathbf {u} \rangle

is the matrix with entries

\left(\langle \mathbf {v} ,\mathbf {u} \rangle \right)_{i,j}=\left\langle v_{i},u_{j}\right\rangle .

${\tilde {u}}_{i}:=(I-P)u_{i},$ and ${\tilde {v}}_{i}:=(I-P)^{*}v_{i}$ then is a biorthogonal system.

See also edit

Dual basis – Linear algebra concept
Dual space – In mathematics, vector space of linear forms
Dual pair
Orthogonality – Various meanings of the terms
Orthogonalization

References edit

^ 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]