From Wikipedia, the free encyclopedia
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.
Projection[edit]
Related to a biorthogonal system is the projection
where
its image is the
linear span of
and the
kernel is
Construction[edit]
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.
See also[edit]
References[edit]
- Jean Dieudonné, On biorthogonal systems Michigan Math. J. 2 (1953), no. 1, 7–20 [1]
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