Biorthogonal system
In mathematics, a biorthogonal system is a pair of indexed families of vectors v ~ i in E and u ~ i in F {\displaystyle {\tilde {v}}_{i}{\text{ in }}E{\text{ and }}{\tilde {u}}_{i}{\text{ in }}F} such that ⟨ v ~ i , u ~ j ⟩ = δ i , j , {\displaystyle \left\langle {\tilde {v}}_{i},{\tilde {u}}_{j}\right\rangle =\delta _{i,j},} where E {\displaystyle E} and F {\displaystyle F} form a pair of topological vector spaces that are in duality, ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \,\cdot ,\cdot \,\rangle } is a bilinear mapping and δ i , j {\displaystyle \delta _{i,j}} 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.