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biorthogonal system : ウィキペディア英語版
biorthogonal system
In mathematics, a biorthogonal system is a pair of indexed families of vectors
:\tilde v_i in and \tilde u_i in
such that
: \langle\tilde v_i , \tilde u_j\rangle = \delta_ ,
where ''E'' and ''F'' form a pair of topological vector spaces that are in duality, is a bilinear mapping and \delta_ is the Kronecker delta.
A biorthogonal system in which and \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.
==Projection==
Related to a biorthogonal system is the projection
:P:= \sum_ \tilde u_i \otimes \tilde v_i ,
where \left( u \otimes v\right) (x):= u \langle v, x\rangle; its image is the linear span of \, and the
kernel is \.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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