Biorthogonal system (original) (raw)

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In mathematics, a biorthogonal system is a pair of indexed families of vectors v ~ i in E and u ~ i in F {\displaystyle {\tilde {v}}_{i}{\text{ in }}E{\text{ and }}{\tilde {u}}_{i}{\text{ in }}F} {\displaystyle {\tilde {v}}_{i}{\text{ in }}E{\text{ and }}{\tilde {u}}_{i}{\text{ in }}F}such that ⟨ v ~ i , u ~ j ⟩ = δ i , j , {\displaystyle \left\langle {\tilde {v}}_{i},{\tilde {u}}_{j}\right\rangle =\delta _{i,j},} {\displaystyle \left\langle {\tilde {v}}_{i},{\tilde {u}}_{j}\right\rangle =\delta _{i,j},}where E {\displaystyle E} {\displaystyle E} and F {\displaystyle F} {\displaystyle F} form a pair of topological vector spaces that are in duality, ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \,\cdot ,\cdot \,\rangle } {\displaystyle \langle \,\cdot ,\cdot \,\rangle } is a bilinear mapping and δ i , j {\displaystyle \delta _{i,j}} {\displaystyle \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 {\displaystyle E=F} {\displaystyle E=F} and v ~ i = u ~ i {\displaystyle {\tilde {v}}_{i}={\tilde {u}}_{i}} {\displaystyle {\tilde {v}}_{i}={\tilde {u}}_{i}} is an orthonormal system.

Related to a biorthogonal system is the projection P := ∑ i ∈ I u ~ i ⊗ v ~ i , {\displaystyle P:=\sum _{i\in I}{\tilde {u}}_{i}\otimes {\tilde {v}}_{i},} {\displaystyle P:=\sum _{i\in I}{\tilde {u}}_{i}\otimes {\tilde {v}}_{i},}where ( u ⊗ v ) ( x ) := u ⟨ v , x ⟩ ; {\displaystyle (u\otimes v)(x):=u\langle v,x\rangle ;} {\displaystyle (u\otimes v)(x):=u\langle v,x\rangle ;} its image is the linear span of { u ~ i : i ∈ I } , {\displaystyle \left\{{\tilde {u}}_{i}:i\in I\right\},} {\displaystyle \left\{{\tilde {u}}_{i}:i\in I\right\},} and the kernel is { ⟨ v ~ i , ⋅ ⟩ = 0 : i ∈ I } . {\displaystyle \left\{\left\langle {\tilde {v}}_{i},\cdot \right\rangle =0:i\in I\right\}.} {\displaystyle \left\{\left\langle {\tilde {v}}_{i},\cdot \right\rangle =0:i\in I\right\}.}

Given a possibly non-orthogonal set of vectors u = ( u i ) {\displaystyle \mathbf {u} =\left(u_{i}\right)} {\displaystyle \mathbf {u} =\left(u_{i}\right)} and v = ( v i ) {\displaystyle \mathbf {v} =\left(v_{i}\right)} {\displaystyle \mathbf {v} =\left(v_{i}\right)} the projection related is P = ∑ i , j u i ( ⟨ v , u ⟩ − 1 ) j , i ⊗ v j , {\displaystyle P=\sum _{i,j}u_{i}\left(\langle \mathbf {v} ,\mathbf {u} \rangle ^{-1}\right)_{j,i}\otimes v_{j},} {\displaystyle P=\sum _{i,j}u_{i}\left(\langle \mathbf {v} ,\mathbf {u} \rangle ^{-1}\right)_{j,i}\otimes v_{j},}where ⟨ v , u ⟩ {\displaystyle \langle \mathbf {v} ,\mathbf {u} \rangle } {\displaystyle \langle \mathbf {v} ,\mathbf {u} \rangle } is the matrix with entries ( ⟨ v , u ⟩ ) i , j = ⟨ v i , u j ⟩ . {\displaystyle \left(\langle \mathbf {v} ,\mathbf {u} \rangle \right)_{i,j}=\left\langle v_{i},u_{j}\right\rangle .} {\displaystyle \left(\langle \mathbf {v} ,\mathbf {u} \rangle \right)_{i,j}=\left\langle v_{i},u_{j}\right\rangle .}

  1. ^ Bhushan, Datta, Kanti (2008). Matrix And Linear Algebra, Edition 2: AIDED WITH MATLAB. PHI Learning Pvt. Ltd. p. 239. ISBN 9788120336186.{{[cite book](/wiki/Template:Cite%5Fbook "Template:Cite book")}}: CS1 maint: multiple names: authors list (link)