The best possible lower bound for the Perron root using traces Original Research Article

Let A be an n×n matrix with real eigenvalues. Wolkowicz and Styan presented bounds for the eigenvalues, using only n, tr A, and tr A2. We show that their lower bound for the largest eigenvalue works also as a lower bound for the Perron root of A if A is nonnegative and its eigenvalues are not necessarily real. We also show that this bound is optimal under certain conditions. Finally, we solve completely the problem to find the optimal lower bound for the Perron root using only n, tr A, and tr A2.