Sankhya: The Indian Journal of Statistics

2002, Volume 64, Series A, Pt. 2, 453--507

**SOME COMMENTS ON SEVERAL MATRIX INEQUALITIES WITH APPLICATIONS TO CANONICAL CORRELATIONS: HISTORICAL BACKGROUND AND RECENT DEVELOPMENTS**

By

S.W. DRURY, McGill University, Montr\'eal, Canada SHUANGZHE LIU, Australian National University, Canberra, Australia CHANG-YU LU, East China Normal University,Shanghai, China SIMO PUNTANEN, University of Tampere, Tampere, Finland and GEORGE P.H. STYAN, McGill University, Montr\'eal, Canada

SUMMARY. We review several matrix inequalities and give some statistical applications, with special emphasis on canonical correlations; many historical and biographical remarks are also included as well as over 100 references. Our paper builds upon the recent survey by Alpargu and Styan (2000) and concentrates on recent developments. We present a new Generalized Matrix Frucht--Kantorovich inequality and show that it is ``essentially equivalent" to the Generalized Matrix Wielandt inequality given by Lu (1999), extending recent results by Wang and Ip (1999). We discuss an interesting special case involving block rank additivity of a partitioned matrix and offer several characterizations. We also consider the Krasnosel$^\prime$ski{\u \i}--Kre\u\i n inequality and the Shisha--Mond inequality and matrix extensions due to Khatri and Rao (1981, 1982) and Rao (1985).

*AMS (1991) subject classification}. *Primary 15A42; secondary 62H20, 62J05.

*Key words and phrases: *Antieigenvalues, block rank additivity, Bloomfield--Watson--Knott inequality, canonical correlations, determinantal inequalities, efficiency of ordinary least squares, Frucht--Kantorovich inequality, generalized correlation coefficient, Khatri--Rao inequality, \KK\ inequality, Kronecker sum, Lyapunov condition, majorization of eigenvalues, measures of association, Rao inequality, %Schur complements, Shisha--Mond inequality, trace inequalities, vector coefficient of alienation, %vector correlation coefficient, Watson efficiency, Wielandt inequality.