Neha Patil (Editor)

Biorthogonal system

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

Contents

v ~ i in E and u ~ i in F

such that

⟨ v ~ i , u ~ j ⟩ = δ i , j ,

where E and F form a pair of topological vector spaces that are in duality, ⟨,⟩ is a bilinear mapping and δ i , j is the Kronecker delta.

A biorthogonal system in which E = F and v ~ i = u ~ i is an orthonormal system.

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

Projection

Related to a biorthogonal system is the projection

P := ∑ i ∈ I u ~ i ⊗ v ~ i ,

where ( u ⊗ v ) ( x ) := u ⟨ v , x ⟩ ; its image is the linear span of { u ~ i : i ∈ I } , and the kernel is { ⟨ v ~ i , ⋅ ⟩ = 0 : i ∈ I } .

Construction

Given a possibly non-orthogonal set of vectors u = ( u i ) and v = ( v i ) the projection related is

P = ∑ i , j u i ( ⟨ v , u ⟩ − 1 ) j , i ⊗ v j ,

where ⟨ v , u ⟩ is the matrix with entries ( ⟨ v , u ⟩ ) i , j = ⟨ v i , u j ⟩ .

  • u ~ i := ( I − P ) u i , and v ~ i := ( I − P ) ∗ v i then is a biorthogonal system.

    • References

    Biorthogonal system Wikipedia
    (Text) CC BY-SA


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