关于矩阵展形的一些新上界

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.

On Some New Upper Bounds for the Spread of a Matrix

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.