Biorthogonal system

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

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

such that

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

where E and F form a pair of topological vector spaces that are in duality, ⟨·,·⟩ is a bilinear mapping and ${\displaystyle \delta _{i,j}}$ 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 and ${\displaystyle {\tilde {v}}_{i}={\tilde {u}}_{i}}$ is an orthonormal system.

Projection

Related to a biorthogonal system is the projection

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

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

Construction

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

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

where ${\displaystyle \langle \mathbf {v} ,\mathbf {u} \rangle }$ is the matrix with entries ${\displaystyle \left(\langle \mathbf {v} ,\mathbf {u} \rangle \right)_{i,j}=\left\langle v_{i},u_{j}\right\rangle }$.

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

See also

• Dual basis
• Dual space
• Dual pair
• Orthogonality
• Orthogonalization

References

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

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