Abstract: | Kantorovich gave an upper bound to the product of two quadratic forms, (X′AX) (X′A−1X), where X is an n-vector of unit length and A is a positive definite matrix. Bloomfield, Watson and Knott found the bound for the product of determinants |X′AX| |X′A−1X| where X is n × k matrix such that X′X = Ik. In this paper we determine the bounds for the traces and determinants of matrices of the type X′AYY′A−1X, X′B2X(X′BCX)−1 X′C2X(X′BCX)−1 where X and Y are n × k matrices such that X′X = Y′Y = Ik and A, B, C are given matrices satisfying some conditions. The results are applied to the least squares theory of estimation. |