非负矩阵的扰动特征值
Perturbing eigenvalues of non-negative matrices
摘要：
令$A$是具有特征值$$ρ, b+ic,b-ic, λ_4,\cdots,λ_n,$$的不可约（按条目）非负$n\times n$矩阵，其中$ρ$是Perron特征值。证明了对于任何$t \in [0, \infty)$，当$\tilde t \ge γ_n t$具有$n \ge 6$的$γ_3=1, γ_4 = 2, γ_5=\sqrt 5$和$γ_n = 2.25$时，存在一个具有特征值$$ρ+ \tilde t,λ_2+t,λ_3+t, λ_4 \cdots,λ_n,$$的非负矩阵.我们的证明依赖于几何学中的一个辅助结果，即边为$n$的凸多边形的面积以$γ_n$乘以位于多边形内部的三角形的最大面积为界.
英文摘要：
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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