F 型广义Z -矩阵与M -矩阵的几个性质

辽宁石油化工大学学报 ›› 2005, Vol. 25 ›› Issue (2): 92-94.

F 型广义Z -矩阵与M -矩阵的几个性质

冯志鑫, 李　阳, 宋岱才

辽宁石油化工大学理学院, 辽宁抚顺113001

收稿日期:2004-09-30 出版日期:2005-06-20 发布日期:2017-04-18

Proper ties of F Class Generalized Z -Matrices and M -Matrices

School of Science , Liaoning University of Petroleum & Chemical Technology ,
Fushun Liaoning 113001, P .R .China

Received:2004-09-30 Published:2005-06-20 Online:2017-04-18

摘要/Abstract

摘要： 　定义了一种新型广义Z -矩阵和广义M -矩阵, 并给出了几个F 型广义Z -矩阵和F 型广义M -矩阵的重要性质。F 型广义M-矩阵不仅包括了M-矩阵, 还包括了所有的正矩阵。若非对角元是非正的, 则矩阵A∈ R ^{n ×n}称为Z -矩阵。当且仅当A 是Z -矩阵同时也是P -矩阵时, A∈ R^{n ×n}称为M -矩阵。对一个方阵进行均分块, 若所有的小方块都是Z -矩阵, 则称此方阵为F 型广义Z -矩阵。对一个方阵进行均分块, 若所有的小块都是M-矩阵, 则称此方阵为F 型广义M -矩阵。得到了F 型广义M-矩阵的一些性质。若M , N ∈ R^{n ×n}皆为相同分类F 型广义M -矩阵, 则在广义FAN 积定义下, M ＊N仍为一个该分类的F 型广义M -矩阵。任意一个F 型广义M -矩阵只有唯一的分法使它成为F 型广义M -矩阵。这些性质为更好的解广义线性互补问题奠定了一定的基础。

关键词: F 型广义M -矩阵, 　广义FAN 积, 　Z -矩阵\　均分块

Abstract:

A new kind of generalized Z - matrices and generalized M - matrices were defined and some properties of the matrices were given .F class generalized M- matrices include not only all M -matrices but also all positive matrices.A matrix A ∈ R^{n ×n} is a Z - matrix if all its non -diagonal elements are non -positive .A ∈ R^{n ×n} is a M- matrix if and only if the A matrix is a Z -matrix and P - matrix .If a square matrix is mean divided and all divided squares are Z - matrices, then it is called F class generalized Z - matrix .If all divided squares are M - matrices , then it is called F class generalized M- matrix .Some proper ties of F class generalized M - matrices were studied too .For example , M , N ∈ R^{n ×n} are the same class of F class generalized M -matrices, under the generalized FAN product , M ＊N is still the class of F class generalized M - matrices.Any F class generalized M- matrix has only one mean division to make it become a F class generalized M -matrix .The results lay a foundation for solving generalized linear complementarity problems

Key words: F class generalized M - matrices, Generalized FAN product , Z - matrices, Mean division

冯志鑫, 李　阳, 宋岱才. F 型广义Z -矩阵与M -矩阵的几个性质[J]. 辽宁石油化工大学学报, 2005, 25(2): 92-94.

FENG Zhi -xin,LI Yang,SONG Dai -cai. Proper ties of F Class Generalized Z -Matrices and M -Matrices [J]. Journal of Liaoning Petrochemical University, 2005, 25(2): 92-94.