A lemma on G-inverse of a matrix and computation of correlation coefficients in the singular case

Abstract

Formulae for multiple, partial and canonical correlation coefficients are generally expressed in terms of the elements of inverse covariance matrix. They are not applicable when the covariance matrix is singular. In this paper, a unified approach is presented to cover the singular and non-singular cases. The formulae involve g-inverse of singular matrices and the results are derived from a lemma on the structure of the idempotent matrix AA_ where A_ is any g-inverse (i.e., AA_A=A). The conditions under which some columns of AA_ are unit vectors are obtained. The formulae for canonical correlations and the canonical transformations of two sets of variables in the singular case are shown to be of the same form as in the non-singular case, with the convention that only the proper eigen values and vectors of determinantal equations are considered.