Perturbing eigenvalues of non-negative matrices

Let $A$ be an irreducible (entrywise) nonnegative $n\times n$ matrix with eigenvalues $ρ, b+ic,b-ic, λ_4,\cdots,λ_n,$ where $ρ$ is the Perron eigenvalue. It is shown that for any $t \in [0, \infty)$ there is a nonnegative matrix with eigenvalues $ρ+ \tilde t,λ_2+t,λ_3+t, λ_4 \cdots,λ_n,$ whenever $\tilde t \ge γ_n t$ with $γ_3=1, γ_4 = 2, γ_5=\sqrt 5$ and $γ_n = 2.25$ for $n \ge 6$. 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 the triangle lying inside the polygon.

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