交矩阵与并矩阵的正定性与特征值
On the positive definiteness and eigenvalues of meet and join matrices
摘要：
本文用一种新的方法研究了交与联矩阵的正定性。当集合$S_n$满足闭时，给出了矩阵$(S_n)_f$正定的一个充必要条件.作为推论，我们从这个条件得到了正定性的一些充分条件。我们还使用图论，并表明，通过一些图论的假设上设置$S_n$，我们能够减少对功能$f$的假设，同时仍然保持矩阵的正定性$(S_n)_f$。并给出了这些结果在连接矩阵上的对偶定理。作为例子，我们考虑所谓的功率GCD和功率LCM矩阵以及MIN和MAX矩阵。最后，当函数$f$具有某种单调性时，给出了交并矩阵特征值的界。
英文摘要：
In this paper we study the positive definiteness of meet and join matrices using a novel approach. When the set $S_n$ is meet closed, we give a sufficient and necessary condition for the positive definiteness of the matrix $(S_n)_f$. From this condition we obtain some sufficient conditions for positive definiteness as corollaries. We also use graph theory and show that by making some graph theoretic assumptions on the set $S_n$ we are able to reduce the assumptions on the function $f$ while still preserving the positive definiteness of the matrix $(S_n)_f$. Dual theorems of these results for join matrices are also presented. As examples we consider the so-called power GCD and power LCM matrices as well as MIN and MAX matrices. Finally we give bounds for the eigenvalues of meet and join matrices in cases when the function $f$ possesses certain monotonic behaviour.
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。
