In mathematics, a biorthogonal system is a pair of indexed families of vectors
- in E and in F
where E and F form a pair of topological vector spaces that are in duality, ⟨·,·⟩ is a bilinear mapping and 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 E = F and is an orthonormal system.
Related to a biorthogonal system is the projection
where ; its image is the linear span of , and the kernel is .
Given a possibly non-orthogonal set of vectors and the projection related is
where is the matrix with entries .
- , and then is a biorthogonal system.
- Jean Dieudonné, On biorthogonal systems Michigan Math. J. 2 (1953), no. 1, 7–20