AI 中文总结
本文研究非循环矩阵逆的结构，发现对角占优非循环矩阵逆存在隐藏衰减，给出其逆元素绝对值的界，相关结果推广了三对角矩阵的对应结论。
AI 中文摘要
众所周知，对角占优三对角矩阵的逆矩阵元素的绝对值会沿对角线向外衰减，且这类绝对值的上下界已被知晓很久。三对角矩阵的图可视为一条线，因此三对角矩阵是特殊的非循环矩阵，即其图为树的矩阵。本文探究非循环矩阵逆的结构，发现对角占优非循环矩阵的逆不存在从对角线开始的严格衰减，该衰减是隐藏的，我们识别出了这种可被清晰描述的衰减，还给出了非循环矩阵逆元素绝对值的界，所有结果均推广了三对角矩阵的相关结论。
英文摘要
It is well-known that the absolute values of the entries of the inverse of a diagonally dominant tridiagonal matrix decay away from the diagonal. Moreover, lower and upper bounds for these absolute values are known for a long time. The graph of a tridiagonal matrix can be viewed as a line. Thus tridiagonal matrices are special acyclic matrices, i.e. matrices whose graph is a tree. In this paper we explore the structure of inverses of acyclic matrices. However, there is no strict decay from the diagonal in the inverses of diagonally dominant acyclic matrices. The decay is somehow hidden. Here we identify this decay which can be described nicely. Moreover we give bounds for the absolute value of the entries of the inverses of acyclic matrices. All of our results generalize results for tridiagonal matrices.
Comments19 pages
