关于矩阵展形的一些新上界  

On Some New Upper Bounds for the Spread of a Matrix

作　　者：廖文诗[1] 

机构地区：[1]重庆大学数学与统计学院,重庆401331

出　　处：《重庆师范大学学报（自然科学版）》2012年第5期46-49,共4页Journal of Chongqing Normal University:Natural Science

摘　　要：文献[1]提出了矩阵的展形,证明了矩阵展形的一个上界估计式,并且给出了这个不等式取等号的条件,即A是正规矩阵且A的特征值满足条件φ时等号成立。本文探讨矩阵展形的新的上界,证明了一个矩阵展形的上界估计式:s(A)≤2‖A‖2F-tr A2n()2-12‖[A,A*]‖2槡F{}12;然后,利用矩阵展形的估计式得到了一个奇异矩阵的谱半径的上界;最后,还给出了两个关于实展形、虚展形的上界的估计式:sRA()≤‖A‖2F-tr A2n()2-12‖[A,A*]‖2槡F+tr A2n+Re tr A2-2ntr B()2()12,sIA()≤‖A‖2F-tr A2n()2-12‖[A,A*]‖2槡F+tr A2n+Re tr A2-2ntr C()2()12.L.Mirsky proposed the spread of a matrix,proved an upper bound for the spread of a matrix and with equality if and only ifAis normal and its eigenvalues satisfy conditionφ.First,in this paper,we discussed new upper bounds for the spread of a matrix.We formulated an upper bound for the spread of a matrix sA≤2‖A‖2F-tr A2 n2-1 2‖‖2F1 2.Second,we obtained an upper bound for spectral radius of singular matrices by a new upper bound for the spread of matrix.Furthermore,two upper bounds for real spread and imaginary spread was given sRA≤‖A‖2F-tr A2 n2-1 2‖‖2F＋tr A2 n＋Re tr A2-2 ntr B21 2, sIA≤‖A‖2F-tr A2 n2-1 2‖‖2F＋tr A2 n＋Re tr A2-2 ntr C21 2.

关 键 词：矩阵展形 谱半径 正规矩阵 非正规矩阵 上界 

分 类 号：O151.2[理学—数学]