Biorthogonal system

In mathematics, a biorthogonal system is a pair of indexed families of vectors
such that
where "E" and "F" form a pair of topological vector spaces that are in duality, is a bilinear mapping and formula_4 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.
A biorthogonal system in which and formula_5 is an orthonormal system.
Projection.
Related to a biorthogonal system is the projection 
where formula_7; its image is the linear span of formula_8, and the kernel is formula_9. 
Construction.
Given a possibly non-orthogonal set of vectors formula_10 and formula_11 the projection related is 
where formula_13 is the matrix with entries formula_14. 