DISTRIBUTION OF MATRIX EIGENVALUE AND ITS APPLICATION IN NUMERICAL ANALYSIS
Jingwei Liang · 2001
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.