Some generalizations of Kantorovich inequality

Khatri, C. G. ; Rao, Radhakrishna C. (1982) Some generalizations of Kantorovich inequality Sankhya, 44 (1). pp. 91-102. ISSN 0972-7671

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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.

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