矩阵对角占优性的推广及应用

A GENERALIZATION OF DIAGONAL DOMINANCE OF MATRICES AND ITS APPLICATIONS

<正> §1.引言设 A=(a_(ij))_(n×n)为一复矩阵,若有一正向量 d=(d_1,d_2,…,d_n)~T 使得d_i|a_(ij)|≥sum from j≠1 d_j|a_(ij)|,(1)对每一 i∈N={1,2,…,n}都成立,则称 A 为广义对角占优矩阵,记为 A∈D_0~*;如若(1)式中每一不等号都是严格的,则称 A 为广义严格对角占优矩阵,记为 A∈D~*.特别地,当 d=(1,1,…,1)~T 时,A∈D_0~*及 A∈D~*即是通常的对角占优与严格对角占优,分别记作 A∈D_0及 A∈D.利用矩阵的对角占优性质讨论其特征值分布是矩阵论中的重要课题,文献[5]—[10]给出了这方面的重要结果.n 阶实方阵 A 称为 M-矩阵,如果 A具有形式:A=sI-B,s>ρ(B),其中 B 为 n 阶非负方阵,ρ(B)表 B 之谱半径,利用广义严格对角占优的概念,文[1]给出了 M-矩阵的等价表征:若 n

In this paper,first we deduce the bound of the absolute smallest eigenvalue of M-mat-rices,introduce the concept of bidiagonally dominant matrices,and obtain a new characte-rization of M-matrices or generalized strictly diagonally dominant matrices.As applicationswe give the distribution theorems of eigenvalues of some matrices,and generalize these resultsto block matrices,thus having improved and generalized the corresponding results of[5]—[11].