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

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

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

${\displaystyle E}$

${\displaystyle F}$

${\displaystyle \langle \,\cdot ,\cdot \,\rangle }$

${\displaystyle \delta _{i,j))$

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 ${\displaystyle E=F}$ and ${\displaystyle {\tilde {v))_{i}={\tilde {u))_{i))$ is an orthonormal system.

Projection

Related to a biorthogonal system is the projection

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

${\displaystyle (u\otimes v)(x):=u\langle v,x\rangle ;}$

${\displaystyle \left\((\tilde {u))_{i}:i\in I\right\},}$

${\displaystyle \left\{\left\langle {\tilde {v))_{i},\cdot \right\rangle =0:i\in I\right\}.}$

Construction

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

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

${\displaystyle \langle \mathbf {v} ,\mathbf {u} \rangle }$

${\displaystyle \left(\langle \mathbf {v} ,\mathbf {u} \rangle \right)_{i,j}=\left\langle v_{i},u_{j}\right\rangle .}$

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