Some comments on several matrix inequalities with applications to canonical correlations: Historical background and recent developments

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 spe cial case involving block rank additivity of a partitioned matrix and offer several characterizations volving block rank additivity of a partitioned matrix and offer several characterizations. We also consider the Krasnosel'ski?-Kre?n inequality and the Shisha-Mond inequality and matrix extensions due to Khatri and Rao (1981, 1982) and Rao (1985) Some related inequalities involving determinants and traces are also presented. We estab lish some new inequalities and give a proof for an upper bound for the product of canonical correlations stated by Khatri (1982) and Khatri and Rao (1982). In addition, we present a new proof of the Bloomfield-Watson-Knott inequality; the Bloomfield-Watson-Knott, Khatri-Rao and Rao inequalities are identified as essential for exciting new results on majorization of eigenvalues due to Ando (2000, 2001) and Li and Mathias (1999)