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In mathematics, a **biorthogonal system** is a pair of indexed families of vectors
such that
where and 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.^{[1]}

A biorthogonal system in which 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.

- Dual basis – Linear algebra concept
- Dual space – In mathematics, vector space of linear forms
- Dual pair
- Orthogonality – Various meanings of the terms
- Orthogonalization

- Jean Dieudonné,
*On biorthogonal systems*Michigan Math. J. 2 (1953), no. 1, 7–20 [1]