Biorthogonal system

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

{\tilde v}_{i}

in

E

and

{\tilde u}_{i}

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, $⟨\cdot,\cdot⟩$ 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

Related to a biorthogonal system is the projection

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

,

where $\left(u\otimes v\right)(x):=u\langle v,x\rangle$ ; 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

Given a possibly non-orthogonal set of vectors $\mathbf {u} =(u_{i})$ 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 ${\tilde {v}}_{i}:=\left(I-P\right)^{*}v_{i}$ then is a biorthogonal system.

References

^ 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 [1]

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

[1]

Biorthogonal system

Projection

Construction

See also

References