M. A. Huda,Md. Harun-or-Roshid,A. Islam ,Mst. Mumtahinah ,



sensitivity,eigenvalues, perturbations,complex eigenvalues,


The main objective of this paper is to study the sensitivity of eigenvalues in their computational domain under perturbations, and to provide a solid intuition with some numerical example as well as to represent them in graphically. The sensitivity of eigenvalues, estimated by the condition number of the matrix of eigenvectors has been discussed with some numerical example. Here, we have also demonstrated, other approaches imposing some structures on the complex eigenvalues, how this structure affects the perturbed eigenvalues as well as what kind of paths do they follow in the complex plane.


I.Aripirala R. and Syrmos V. L., “Sensitivity Analysis of Stable GeneralizedLyapunov Equations,” In Proc. of the 32nd IEEE Conf. on Decision and Control, pp. 3144-3129, San Antonio, 1993.

II.Bhatia R., Eisner L. and Krause G., “Bounds for the Variation of the Rootsof a Polynomial and the Eigenvalues of a Matrix,” Linear Algebra Appl.,142, 195-209, 1990.

III.Elsner L., “An Optimal Bound for the Spectral Variation of Two Matrices,”Linear” Algebr’a and Its Applications, 71:77 -80, 1985.

IV.Eslami M., “Theory of Sensitivity in Dynamic Systems,” Springer-Verlag,Berlin, 1994.

V.Holbrook J. A. R., “Spectral Variation of Normal Matrices,” LinearAlgbera Appl., 174:131-144, 1994.

VI.J.B. Hiriart-Urruty J.B. and Ye D., “Sensitivity Analysis of All Eigenvalues of aSymmetric Matrix,” Numer. Math., 70:45-72, 1992.

VII.Hewer G. and Kenney C., “The Sensitivity of the Stable LyapunovEquation,” SIAM J. Cont. Optim., 26: 321-344, 1998.

VIII.Ipsen I. C. F., “Relative Perturbation Results for Matrix Eigenvalues andSingular Values,” Acta Numer, 7:151-201, 1998.

IX.Konstanitinov M., Petkov P., GU D. W. and Mehrmann V., “Sensitivity of.General Lyapunov Equations,” Technical report 98-15, Dept. of Engineering, Leicester univ., UK, 1998.

X.Konstantinov M., Petkov P. and Angelova V., “ Sensitivity of GeneralDiscrete Algebraic Riccati Equations,” In Proc. 28 Spring Conf. of Union of Bulgar. Mathematics, pp. 128-136, Bulgaria, 1999.

XI.Moler C. B., Numerical Computing with MATLAB, February 15, 2008.

XII.Ostrowski A., “Dber die Stetigkeit von charakteristischen Wurzeln inAbhiingigkeit von den Matrizenelementen,” Jahresberichte der Deutsche Mathematische Ver”ein 60, 40-42, 1957.

XIII.Parlett B. N., “The Symmetric Eigenvalue Problem,” Prentice-Hall,Englewood Cliffs, NJ, 1980.

XIV.Rump S. M., “Estimation of the Sensitivity of Linear and NonlinearAlgebraic Problems,” Linear algebra, Appl., 153:1-34, 1991.

XV.Rajendra B., “Perturbation Bounds for Matrix Eigenvalues,” SIAM, Wiley,New York, 2007.

XVI.Sun J.G., “On the Perturbation of the Eigenvalues of a Normal Matrix,”Math. Numer. Sinica, 6 334-336, 1984.

XVII.Stewart G. W, Sun J., “Matrix Perturbation Theory,” Academic Press. Inc,New York, 2000.

XVIII.Sun J. G, “Sensitivity Analysis of the Discrete-Time Algebraic RiccatiEquation,” Lin. Alg. Appl., 275-276: 595-615, 1998.

XIX.Wilkinson J. H, “Rounding Errors in Algebraic Processes,” Prentice Hall,Englewood Cliffs, 1963.

XX.Wilkinson J., “The Algebraic Eigenvalue Problem,” Clarendon Press, Oxford, 1965.

XXI.Xu S. “Sensitivity Analysis of the Algebraic Riccati Equations,” Numer.Math., 75: 121-134, 1996.

M. A. Huda, Md. Harun-or-Roshid, A. Islam ,Mst. Mumtahinah View Download