Löwdin's canonical orthogonalization: Getting round the restriction of linear independence

Abstract

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.