INEQUALITIES OF RAYLEIGH QUOTIENTS AND BOUNDS ON THE SPECTRAL-RADIUS OF NONNEGATIVE SYMMETRICAL 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 r oot denotes coordinatewise square roots and circle is the Hadamard pro duct, but that Q(A)(Au) may be smaller than Q(A)(u). Further, we exami ne issues of convergence. (C) 1997 Published by Elsevier Science Inc.