Comparison of Ranks of Cross-Product and Covariance Solutions in Component Analysis

Abstract

Suppose D is a data matrix for N persons and n variables, and \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\dot D$$\end{document} is the matrix obtained from D by expressing the variables in deviation-score form. It is shown that if D has rank r, \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\dot D$$\end{document} will always have rank (r − 1) if r = N < n, otherwise it will generally have rank r. If \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\dot D$$\end{document} has rank s, D will always have rank s if s = n, but if s<n it will generally have rank (s + 1). Thus two cases can arise, Case A in which D has rank one greater than \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\dot D$$\end{document}, and Case B in which D has rank equal to \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\dot D$$\end{document}. Implications of this distinction for analysis of cross products versus analysis of covariances are briefly indicated.