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 ${\displaystyle \delta _{i,j}}$ is the Kronecker delta.

A biorthogonal system in which E = F and ${\displaystyle {\tilde {v}}_{i}={\tilde {u}}_{i}}$ is an orthonormal system.

An example is the pair of sets of respectively left and right eigenvectors of a matrix, indexed by eigenvalue.

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