Perturbing eigenvalues of nonnegative matrices

Abstract Let A be an irreducible (entrywise) nonnegative n × n matrix with eigenvalues ρ , λ 2 = b + i c , λ 3 = b − i c , λ 4 , ⋯ , λ n , where ρ is the Perron eigenvalue. It is shown that for any t ∈ [ 0 , ∞ ) there is a nonnegative matrix with eigenvalues ρ + t ˜ , λ 2 + t , λ 3 + t , λ 4 , ⋯ , λ n , whenever t ˜ ⩾ γ n t with γ 3 = 1 , γ 4 = 2 , γ 5 = 5 and γ n = 2.25 for n ⩾ 6 . The result improves that of Guo et al. Our proof depends on an auxiliary result in geometry asserting that the area of an n -sided convex polygon is bounded by γ n times the maximum area of a triangle lying inside the polygon.