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Löwdin's canonical orthogonalization: Getting round the restriction of linear independence
Author(s) -
Naidu A. Ramesh,
Srivastava Vipin
Publication year - 2004
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.20136
Subject(s) - orthogonalization , curse of dimensionality , constraint (computer aided design) , independence (probability theory) , space (punctuation) , principal component analysis , vector space , set (abstract data type) , mathematics , combinatorics , algorithm , computer science , pure mathematics , statistics , geometry , programming language , operating system
Löwdin's canonical orthogonalization procedure can be useful in organizing large data sets, but it is applicable only to a set of linearly independent vectors. This places a serious constraint for there can be at most n linearly‐independent vectors in an n ‐dimensional space. We propose two ways of getting round this restriction so that Löwdin's procedure can be used to find the vector along which all the given vectors—any number of them in a space of arbitrary dimensionality—project maximally. Under these conditions, this orthogonalization procedure is equivalent to the principal component analysis. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004