First Order Asymptotic Expansions for Eigenvalues of Multiplicatively Perturbed Matrices

Given an arbitrary square matrix $A$, we obtain the leading terms of the asymptotic expansions in the small, real parameter $\varepsilon$ of multiplicative perturbations $\widehat{A}(\varepsilon)=(I+\varepsilon\,B)A(I+\varepsilon\,C)$ of $A$ for arbitrary matrices $B$ and $C$. The analysis is separated into two rather different cases, depending on whether the unperturbed eigenvalue is zero or not. It is shown that in either case the leading exponents are obtained from the partial multiplicities of the eigenvalue of interest, and the leading coefficients generically involve only appropriately normalized left and right eigenvectors of $A$ associated with that eigenvalue, with no need of generalized eigenvectors. Similar results are obtained for multiplicative perturbation of singular values as well.

[1]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[2]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[3]  H. Baumgärtel.  Analytic perturbation theory for matrices and operators , 1985 .

[4]  M. Overton,et al.  On the Lidskii-Vishik-Lyusternik Perturbation Theory for Eigenvalues of Matrices with Arbitrary Jordan Structure , 1997, SIAM J. Matrix Anal. Appl..

[5]  J. Barlow,et al.  Computing accurate eigensystems of scaled diagonally dominant matrices: LAPACK working note No. 7 , 1988 .

[6]  Chi-Kwong Li,et al.  The Lidskii-Mirsky-Wielandt theorem – additive and multiplicative versions , 1999, Numerische Mathematik.

[7]  Ilse C. F. Ipsen,et al.  Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices , 1998 .

[8]  Froilán M. Dopico,et al.  Low Rank Perturbation of Jordan Structure , 2003, SIAM J. Matrix Anal. Appl..

[9]  J. Stillwell,et al.  Plane Algebraic Curves , 1986 .

[10]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[11]  V. Mehrmann,et al.  Eigenvalue perturbation theory of symplectic, orthogonal, and unitary matrices under generic structured rank one perturbations , 2014 .

[12]  Rui Ralha,et al.  Perturbation Splitting for More Accurate Eigenvalues , 2009, SIAM J. Matrix Anal. Appl..

[13]  S. V. Savchenko.  On the Change in the Spectral Properties of a Matrix under Perturbations of Sufficiently Low Rank , 2004 .

[14]  Froilán M. Dopico,et al.  First Order Eigenvalue Perturbation Theory and the Newton Diagram , 2002 .

[15]  V. Lidskii.  Perturbation theory of non-conjugate operators , 1966 .

[16]  Ren-Cang Li.  Relative Perturbation Theory: I. Eigenvalue and Singular Value Variations , 1998, SIAM J. Matrix Anal. Appl..

[17]  M. Vishik,et al.  THE SOLUTION OF SOME PERTURBATION PROBLEMS FOR MATRICES AND SELFADJOINT OR NON-SELFADJOINT DIFFERENTIAL EQUATIONS I , 1960 .

[18]  V. Puiseux.  Recherches sur les fonctions algébriques. , 1850 .

[19]  A. Ostrowski.  A QUANTITATIVE FORMULATION OF SYLVESTER'S LAW OF INERTIA. , 1959, Proceedings of the National Academy of Sciences of the United States of America.

[20]  Ilse C. F. Ipsen.  Relative perturbation results for matrix eigenvalues and singular values , 1998, Acta Numerica.

[21]  L. Hörmander,et al.  A Remark on Perturbations of Compact Operators. , 1994 .

[22]  Ren-Cang Li.  Relative perturbation theory. III. More bounds on eigenvalue variation , 1997 .