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In [[mathematics]], a '''biorthogonal system''' is a pair of [[indexed family|indexed families]] of vectors
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:<math>\tilde v_i</math> in {{mvar|E}} and <math>\tilde u_i</math> in {{mvar|F}}
such that
:<math> \langle\tilde v_i , \tilde u_j\rangle = \delta_{i,j} ,</math>
where ''E'' and ''F'' form a pair of [[topological vector space]]s that are in [[dual space|duality]], {{math|{{langle}},{{rangle}}}} is a [[bilinear mapping]] and <math>\delta_{i,j}</math> is the [[Kronecker delta]].


A biorthogonal system in which {{math|1={{var|E}} = {{var|F}}}} and <math>\tilde v_i = \tilde u_i</math> is an [[Orthogonal basis|orthonormal system]].
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An example is the pair of sets of respectively left and right eigenvectors of a matrix, indexed by eigenvalue.{{cn|date=January 2013}}
 
==Projection==
Related to a biorthogonal system is the projection
:<math>P:= \sum_{i \in I} \tilde u_i \otimes \tilde v_i </math>,
where <math>\left( u \otimes v\right) (x):= u \langle v, x\rangle</math>; its image is the [[linear span]] of <math>\{\tilde u_i: i \in I\}</math>, and the
[[kernel (algebra)|kernel]] is <math>\{\langle\tilde v_i, \cdot\rangle = 0: i \in I  \}</math>.  
 
==Construction==
Given a possibly non-orthogonal set of vectors <math>\mathbf{u}= (u_i)</math> and <math>\mathbf{v}= (v_i)</math> the projection related is
:<math>P= \sum_{i,j} u_i \left( \langle\mathbf{v}, \mathbf{u}\rangle^{-1}\right)_{j,i} \otimes v_j</math>,
where <math> \langle\mathbf{v},\mathbf{u}\rangle </math> is the matrix with entries <math> \left(\langle\mathbf{v},\mathbf{u}\rangle\right)_{i,j}= \langle v_i, u_j\rangle </math>.
* <math>\tilde u_i:= (I-P) u_i</math>, and <math>\tilde v_i:= \left(I-P \right)^* v_i</math> then is an orthogonal system.
 
==See also==
* [[Dual space]]
* [[Dual pair]]
* [[Orthogonality]]
* [[Orthogonalization]]
 
==References==
 
*Jean Dieudonné, ''On biorthogonal systems'' Michigan Math. J.  2 (1953), no. 1, 7–20  [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.mmj/1028989861]
 
[[Category:Topological vector spaces]]

Revision as of 22:43, 4 February 2014

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