Some generalizations of Kantorovich inequality

Abstract

Kantorovich gave an upper bound for the product (x'Vx)(x'V^-1x) where x is an n-vector of unit length and V is an nXn positive definite matrix. Bloomfield, Watson and Knott found the bound to |X'VXX'V^-1X|, and we found bounds for the trace and determinant of X'VYY'V^-1-1X where X and Y are nXk matrices such that X'X=Y'Y=I. In the present paper we establish bounds for traces and determinants of X'VYY'V^-1-1X and X'BYY'CX when X and Y are matrices of different orders. A review of previous results on generalizations of the Kantorovich inequality and a number of new results of independent interest are also given.