发表机构
University of Regina; Washington State University（里贾纳大学; 华盛顿州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文系统考察方阵正性在整数幂和分数矩阵根下的封闭性，为多类重要矩阵证明正性在分数幂下保持，并强调Pick函数的作用，提供证明、反例及汇总表格。
AI 中文摘要
标量正性概念到方阵的自然推广形式多种多样。文中列表和表格展示了我们在此考虑的推广形式；它们几乎穷尽了实践和理论中出现的情况。由于正标量在任意幂和根运算下保持封闭，自然要问这些矩阵推广形式中哪些具有类似的封闭性质。本文旨在考察并推进这些推广形式在整数幂和分数矩阵根下的封闭性。除综述已知结果外，我们为几类重要矩阵建立了正性在分数幂下保持的性质，表明关键结构特征在矩阵根运算下得以保留。Pick函数在分析中的作用贯穿全文。文中给出了证明、反例及相关结果，并附有汇总表格，以提供简明实用的参考。
英文摘要
There is a remarkable variety of natural generalizations of the notion of scalar positivity to square matrices. The lists and tables indicate those that we consider here; they are practically exhaustive of those that arise in practice and theory. Since positive scalars are closed under taking arbitrary powers and roots, it is natural to ask which of these matrix generalizations enjoy analogous closure properties. Our purpose here is to examine and advance these generalizations with respect to their closure under integer powers and fractional matrix roots. In addition to surveying known results, we establish for several important classes of matrices that positivity is preserved under fractional powers, showing that key structural features are retained under matrix roots. The role of Pick functions in the analysis is emphasized throughout. Proofs, counterexamples, and related results are presented, and summary tables are included to provide a concise and useful reference.
