# 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, ⟨·,·⟩ 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.

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.