Biorthogonal system

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

v ~

i

in

u ~

in

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

such that

v ~

i

,

u ~

δ

and

form a pair of topological vector spaces that are in duality,

⟨

is a bilinear mapping and

δ

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

is an orthonormal system.

Related to a biorthogonal system is the projection

its image is the linear span of

and the kernel is

Given a possibly non-orthogonal set of vectors

the projection related is

is the matrix with entries