Biorthogonal system

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

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

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

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

• Dual basis – Linear algebra concept
• Dual space – In mathematics, vector space of linear forms
• Dual pair – Dual pair of vector spacesPages displaying short descriptions of redirect targets
• Orthogonality – Various meanings of the terms
• Orthogonalization – Process in linear algebra

• Jean Dieudonné, On biorthogonal systems Michigan Math. J. 2 (1953), no. 1, 7–20 [1]

• Topological vector spaces

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