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 . 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.