矩阵特征值的分布及其在数值分析中的应用

DISTRIBUTION OF MATRIX EIGENVALUE AND ITS APPLICATION IN NUMERICAL ANALYSIS

对于n阶复方阵A ,其所有特征值都位于如下的单一圆盘中 : D :z :z - trAn ≤R1=n - 12n - 1n - 1n q +q2 - 2n - 1n2 ΔA ,且这些特征值的实部和虚部分别位于如下的区间 : trReAn - n - 1n qRe, trReAn + n - 1n qRe , trImAn - n - 1n qIm , trImAn + n - 1n qIm .其中 ,q =A 2 F - 1n trA 2 ,ΔA =12 AA -A A 2 F,qRe=ReA 2 F - 1n(trReA ) 2 ,qIm =ImA 2 F - 1n(trImA) 2 .同时 。

It is proved that all eigenvalues for any n -square complex matrix A lie in a single disc as follows: D:JB({z:JB(|z-SX( tr AnSX)JB)|≤R 1=KF(SX(n-12n-1SX)KF)KF(SX(n-1nSX)q+KF(q 2-SX(2n-1n 2SX)Δ AKF)KF)JB)}, and the real and imaginary parts of these eigenvalues lie in the following intervals respectively: JB([SX(trRe AnSX)-KF(SX(n-1nSX)q ReKF), SX(trRe AnSX)+KF(SX(n-1nSX)q ReKF)JB)], JB([SX(trIm AnSX)-KF(SX(n-1nSX)q ImKF), SX(trIm AnSX)+KF(SX(n-1nSX)q ImKF)JB)]. where, q=JB(=AJB)= 2 F-SX(1nSX)JB(| tr AJB)| 2, Δ A=SX(12SX)JB(=AA *-A *AJB)= 2 F,q Re =JB(= Re AJB)= 2 F-SX(1nSX)( trRe A) 2,q Im =JB(= Im AJB)= 2 F-SX(1nSX)( trIm A) 2.At the same time, on the basis of these results, some applications in the estimations for the spectral radius of Jacobi iterative matrix and the optimum relexation parameters of a system of linear equations are also obtained.