A necessary condition for the spectrum of nonnegative symmetric 5 × 5 matrices

Let A be a nonnegative symmetric 5×5 matrix with eigenvalues λ1≥λ2≥λ3≥λ4≥λ5. We show that if ∑i=15λi≥12λ1 then λ3≤∑i=15λi. McDonald and Neumann showed that λ1+λ3+λ4≥0. Let σ=(λ1,λ2,λ3,λ4,λ5) be a list of decreasing real numbers satisfying: