Inequalities of Rayleigh quotients and bounds on the spectral radius of nonnegative symmetric matrices

Given a square, nonnegative, symmetric matrix A, the Rayleigh quotient of a nonnegative vector u under A is given by Q(A)(u) = u(T)Au/u(T)u. We show that Q(A)(root u circle Au) is not less than Q(A)(u), where root denotes coordinatewise square roots and circle is the Hadamard product, but that Q(A)(Au) may be smaller than Q(A)(u). Further, we examine issues of convergence. (C) 1997 Published by Elsevier Science Inc.