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Biorthogonal system

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ARTICLE ORIGINS Biorthogonal system
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In mathematics, a biorthogonal system is a pair of indexed families of vectors
tilde v_i in {{mvar|E}} and tilde u_i in {{mvar|F}}
such that
leftlangletilde v_i , tilde u_jrightrangle = delta_{i,j} ,
where E and F form a pair of topological vector spaces that are in duality, {{math|Â·,Â·{{rangle}}}} is a bilinear mapping and 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.BOOK, Bhushan, Datta, Kanti, Matrix And Linear Algebra, Edition 2: AIDED WITH MATLAB, 2008, PHI Learning Pvt. Ltd., 9788120336186, 239,weblink en, A biorthogonal system in which {{math|1={{var|E}} = {{var|F}}}} and tilde v_i = tilde u_i is an orthonormal system.

Projection

Related to a biorthogonal system is the projection
P := sum_{i in I} tilde u_i otimes tilde v_i ,
where left(u otimes vright) (x) := u langle v, xrangle; its image is the linear span of left{tilde u_i: i in Iright}, and the kernel is left{leftlangletilde v_i, cdotrightrangle = 0: i in Iright}.

Construction

Given a possibly non-orthogonal set of vectors mathbf{u} = (u_i) and mathbf{v} = left(v_iright) the projection related is
P = sum_{i,j} u_i left( langlemathbf{v}, mathbf{u}rangle^{-1}right)_{j,i} otimes v_j,
where langlemathbf{v},mathbf{u}rangle is the matrix with entries left(langlemathbf{v},mathbf{u}rangleright)_{i,j} = leftlangle v_i, u_jrightrangle .
• tilde u_i:= (I - P) u_i, and tilde v_i:= left(I - Pright)^ v_i then is a biorthogonal system.

References

{{reflist}}
• Jean DieudonnÃ©, On biorthogonal systems Michigan Math. J. 2 (1953), no. 1, 7â€“20 weblink

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