齐次单调函数的Perron-Frobenius定理
The Perron-Frobenius Theorem for Homogeneous, Monotone Functions
摘要：
如果A是一个非负矩阵，其关联的有向图是强连通的，则Perron-Frobenius定理断言A在正锥（R^+）^n中有一个特征向量。我们将有向图与任何齐次、单调函数f联系起来：（R^+）^n ->（R^+）^n，并证明如果该图是强连通的，则f在（R^+）^n中有一个（非线性）特征向量。文献中的几个结果作为推论出现。我们的方法表明，Perron-Frobenius定理是“真正的”关于不变子集的有界性在希尔伯特射影度量。它们导致进一步的存在性结果和开放性问题。
英文摘要：
If A is a nonnegative matrix whose associated directed graph is strongly connected, the Perron-Frobenius theorem asserts that A has an eigenvector in the positive cone, (R^+)^n. We associate a directed graph to any homogeneous, monotone function, f: (R^+)^n -> (R^+)^n, and show that if the graph is strongly connected then f has a (nonlinear) eigenvector in (R^+)^n. Several results in the literature emerge as corollaries. Our methods show that the Perron-Frobenius theorem is ``really'' about the boundedness of invariant subsets in the Hilbert projective metric. They lead to further existence results and open problems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
