Comparison of ranks of cross-product and covariance solutions in component analysis

Abstract

SupposeD is a data matrix forN persons andn variables, and$\dot D$ is the matrix obtained fromD by expressing the variables in deviation-score form. It is shown that ifD has rankr,$\dot D$ will always have rank (r−1) ifr=N<n, otherwise it will generally have rankr. If$\dot D$ has ranks,D will always have ranks ifs=n, but ifs<n it will generally have rank (s+1). Thus two cases can arise, Case A in whichD has rank one greater than$\dot D$, and Case B in whichD has rank equal to$\dot D$. Implications of this distinction for analysis of cross products versus analysis of covariances are briefly indicated.