Inequalities of Rayleigh quotients and bounds on the spectral radius of nonnegative symmetric matrices (Q1368768)

Let \(A\) be a nonnegative symmetric matrix, \(u\) be a nonnegative vector. \(Q_A(u)= u^TAu/ u^Tu\) is called the Rayleigh quotient of \(u\) under \(A\). The following result is obtained: \(Q_A (\sqrt{u\circ Au})\leq Q_A(u)\), where \(\sqrt{\;}\) denotes coordinatewise square roots and \(\circ\) is the Hadamard product. \(Q_A(Au)\) may be smaller than \(Q_A(u)\).