SUPPORT THE WORK

GetWiki

Biorthogonal system

ARTICLE SUBJECTS
aesthetics  →
being  →
complexity  →
database  →
enterprise  →
ethics  →
fiction  →
history  →
internet  →
knowledge  →
language  →
licensing  →
linux  →
logic  →
method  →
news  →
perception  →
philosophy  →
policy  →
purpose  →
religion  →
science  →
sociology  →
software  →
truth  →
unix  →
wiki  →
ARTICLE TYPES
essay  →
feed  →
help  →
system  →
wiki  →
ARTICLE ORIGINS
critical  →
discussion  →
forked  →
imported  →
original  →
Biorthogonal system
[ temporary import ]
please note:
- the content below is remote from Wikipedia
- it has been imported raw for GetWiki
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|·,·{{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.

See also

References

{{reflist}}
  • Jean Dieudonné, On biorthogonal systems Michigan Math. J. 2 (1953), no. 1, 7–20 weblink


- content above as imported from Wikipedia
- "Biorthogonal system" does not exist on GetWiki (yet)
- time: 3:57pm EDT - Mon, Jun 24 2019
[ this remote article is provided by Wikipedia ]
LATEST EDITS [ see all ]
GETWIKI 09 MAY 2016
GETWIKI 18 OCT 2015
M.R.M. Parrott
Biographies
GETWIKI 20 AUG 2014
GETWIKI 19 AUG 2014
GETWIKI 18 AUG 2014
Wikinfo
Culture
CONNECT