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 QA(u) = uTAu/uTu. We show that QA(√u° Au) is not less than QA(u), where √ denotes coordinatewise square roots and ° is the Hadamard product, but that QA(Au) may be smaller than QA(u). Further, we examine issues of convergence. © 1997 Published by Elsevier Science Inc.