矩阵特征值问题的Bendixson定理的改进

任一n×n矩阵A可分解为A=B C,其中B=1/2(A A~H),C=1/2(A-A~H)。Bendixson定理的主要内容是:λ_j(A)(j=1,2,…,n)落在矩形区域F上,而构成F的四个边的直线分别为x=max(λ_j(B)),x=min(λ_j(B)),y=max(-iλ_j(C)),y=min(-iλ_j(C))。本文给出用B,C的特征值和矩阵A的正规性偏离度对A的特征值的进一步估计。

AN IMPROVEMENT ON BENDIXSON'S THEOREM FOR AN EIGENVALUE PROBLEM OF A MATRIX

The Bendixson's theorem shows that all the eigenvalues of a matrix A lie in arectangle of the complex plane with four edges defined by the extreme eigenvalues ofmatrices B=1/2(A+A~H) and C=1/2(A-A~H). In this paper, some more delicate bounds foreach eigenvalue of A, denoted by the eigenvalues of B, C and the departure from norma-lity of the matrix A, have been given.