Perturbation analysis of eigenvalues of polynomial matrices smoothly depending on parameters (original) (raw)

Matrix Polynomials Subjected to Small Perturbations

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1986

Perturbation theory is wed to obtain first order shifts in the latent roots and latent vectors of Lambda-Matrices. Both m e 8 of distinct and multiple roots are handled.

On eigenvalues of perturbed quadratic matrix polynomials

Integral Equations and Operator Theory, 1995

Among other results, it is shown that if C and K are arbitrary complex n x n matrices and if det(1gI+,~oC+K) = 0 for some)'0 r 0 (resp. ,~o = 0), then the Newton diagram of the polynomial t (),, e) = det (,~ 21 + ~ (1 + e) C + K), expanded in (~-)~0) and e, has at least a point on or below t he line x + y-b (resp. has no point on or above the line x = y), where b is the algebraic multiplicity of 0 as an eigenvalue of)~02I + 3,0C + K. These are extensions of similar results due to H. Langer, B. Najman, and K. Veselid proved for diagonable matrices C. and shed light on the eigenvalues of the perturbed quadratic matrix polynomids. Our proofs are independent and seem to be simpler.

Smooth multiparameter perturbation of polynomials and operators

Let P (x)(z) = z n + P n j=1 (−1) j a j (x)z n−j be a family of polynomials of fixed degree n whose coefficients a j are germs at 0 of smooth (C ∞ ) complex valued functions defined near 0 ∈ R q . We show: If P is generic there exists a finite collection T of transformations Ψ : R q , 0 → R q , 0 such that S {im(Ψ) : Ψ ∈ T } is a neighborhood of 0 and, for each Ψ ∈ T , the family P • Ψ allows smooth parameterizations of its roots near 0. Any Ψ ∈ T is a finite composition of linear coordinate changes and transforma-

Quasianalytic multiparameter perturbation of polynomials and normal matrices

Transactions of the American Mathematical Society, 2011

We study the regularity of the roots of multiparameter families of complex univariate monic polynomials P (x)(z) = z n + P n j=1 (−1) j a j (x)z n−j with fixed degree n whose coefficients belong to a certain subring C of C ∞functions. We require that C includes polynomial but excludes flat functions (quasianalyticity) and is closed under composition, derivation, division by a coordinate, and taking the inverse. Examples are quasianalytic Denjoy-Carleman classes, in particular, the class of real analytic functions C ω .

Perturbations of Polynomials with Operator Coefficients

Journal of Complex Analysis, 2013

We consider polynomials whose coefficients are operators belonging to the Schatten-von Neumann ideals of compact operators in a Hilbert space. Bounds for the spectra of perturbed pencils are established. Applications to differential and difference equations are also discussed.

On condition numbers of polynomial eigenvalue problems

Applied Mathematics and Computation, 2010

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition numbers of eigenvalues and the pseudospectral growth rate. We obtain that if a simple eigenvalue of a matrix polynomial is ill-conditioned in some respects, then it is close to be multiple, and we construct an upper bound for this distance (measured in the euclidean norm). We also derive a new expression for the condition number of a simple eigenvalue, which does not involve eigenvectors. Moreover, an Elsner-like perturbation bound for matrix polynomials is presented. j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a m c

Perturbations of functions of diagonalizable matrices

The Electronic Journal of Linear Algebra, 2010

Let A andà be n × n diagonalizable matrices and f be a function defined on their spectra. In the present paper, bounds for the norm of f (A) − f (Ã) are established. Applications to differential equations are also discussed.

On the sensitivities of multiple eigenvalues

We consider the generalized symmetric eigen-value problem where matrices depend smoothly on a parameter. It is well known that in general individual eigen-values, when sorted in accordance with the usual ordering on the real line, do not depend smoothly on the parameter. Nevertheless, symmetric polynomials of a number of eigen-values, regardless of their multiplicity, which are known to be isolated from the rest depend smoothly on the parameter. We present explicit readily computable expressions for their first derivatives. Finally, we demonstrate the utility of our approach on a problem of finding a shape of a vibrating membrane with a smallest perimeter and with prescribed four lowest eigenvalues, only two of which have algebraic multiplicity one.

On the location of eigenvalues of matrix polynomials

Operators and Matrices, 2019

A number λ ∈ C is called an eigenvalue of the matrix polynomial P (z) if there exists a nonzero vector x ∈ C n such that P (λ)x = 0. Note that each finite eigenvalue of P (z) is a zero of the characteristic polynomial det(P (z)). In this paper we establish some (upper and lower) bounds for eigenvalues of matrix polynomials based on the norm of their coefficient matrices and compare these bounds to those given by N.J. Higham and F. Tisseur [8], J. Maroulas and P. Psarrakos [12].

Eigenstructure of of singular systems. Perturbation analysis of simple eigenvalues

International Journal of Mathematical Models and Methods in Applied Sciences

The problem to study small perturbations of simple eigenvalues with a change of parameters is of general interest in applied mathematics. After to introduce a systematic way to know if an eigenvalue of a singular system is simple or not, the aim of this work is to study the behavior of a simple eigenvalue of singular linear system family \left .\begin{array}{rl}E(p)\dot x&=A(p)x+B(p)u,\\ y&=C(p)x\end{array}\right \}$$ smoothly dependent on real parameters p=(p1,ldots,pn)p=(p_{1},\ldots ,p_{n})p=(p1,ldots,pn).

The sensitivity of eigenvalues under elementary matrix perturbations

Linear Algebra and its Applications, 1987

Let M be an n X n real matrix, and let E,, be the elementary matrix with 1 in the (x, y) position and zero elsewhere. For L E C we call the matrix M + .zE,, an elementary matrix perturbation of M. Let X be any eigenvalue of M. Then there exists an (x, y) pair, 1~ x, y < n, and an analytic function h,,(z) defined in a neighborhood N of the origin such that: (a) h,,(O) = X. (b) h,,(z) is an eigenvalue of the elementary matrix perturbation M + .z~(~)E_, for any z E N, where k(X) is the dimension of the largest block containing X in the Jordan canonical form of M. (c) For any z E N, .z # 0, M + zE,, has k(X) distinct eigenvalues, all different from X. If X(z) is any one of these, then 1X -h(z)1 = O(l~li/~(~)). (d) For any z E N, .z # 0, M + zE,, has eigenvalue X with multiplicity s(A) -k(X), where s(X) is the (algebraic) multiplicity of X in M. (e) For all real positive or negative t E N, but

Perturbation results related to palindromic eigenvalue problems

ANZIAM Journal, 2009

We investigate the perturbation of the palindromic eigenvalue problem for the matrix quadratic P (λ) ≡ λ 2 A 1 + λA 0 + A 1 , with A 0 , A 1 ∈ C n×n and A 0 = A 0 (= T, H). The perturbation of eigenvalues, in terms of general matrix polynomials, palindromic pencils, (semi-Schur) anti-triangular canonical forms and differentiation, are discussed.

Stable perturbations of nonsymmetric matrices

Linear Algebra and its Applications, 1992

A complex matrix is said to be stable if all its eigenvalues have negative real part. Let J be a Jordan block with zeros on the diagonal and ones on the superdiagonal, and consider analytic matrix perturbations of the form A() = J + B + O(2), where is real and positive. A necessary condition on B for the stability of A() on an interval (0; 0), and a su cient condition on B for the existence of such a family A(), is (i) Re tr B 0; (ii) the sum of the elements on the rst subdiagonal of B has nonpositive real part and zero imaginary part; (iii) the sum of the elements on each of the other subdiagonals of B is zero. This result is extended to matrices with any number of nonderogatory eigenvalues on the imaginary axis, and to a stability de nition based on the spectral radius. A generalized necessary condition, though not a su cient condition, applies to arbitrary Jordan structures. The proof of our results uses two important techniques: the Puiseux-Newton diagram and the normal form of Arnold. In the nonderogatory case our main results were obtained by Levantovskii in 1980 using a di erent proof. Practical implications are discussed.

Perturbation of complex polynomials and normal operators

Mathematische Nachrichten, 2009

We study the regularity of the roots of complex monic polynomials P (t) of fixed degree depending smoothly on a real parameter t. We prove that each continuous parameterization of the roots of a generic C ∞ curve P (t) (which always exists) is locally absolutely continuous. Generic means that no two of the continuously chosen roots meet of infinite order of flatness. Simple examples show that one cannot expect a better regularity than absolute continuity. This result will follow from the proposition that for any t 0 there exists a positive integer N such that t → P (t 0 ± (t − t 0 ) N ) admits smooth parameterizations of its roots near t 0 . We show that C n curves P (t) (where n = deg P ) admit differentiable roots if and only if the order of contact of the roots is ≥ 1. We give applications to the perturbation theory of normal matrices and unbounded normal operators with compact resolvents and common domain of definition: The eigenvalues and eigenvectors of a generic C ∞ curve of such operators can be arranged locally in an absolutely continuous way.

Perturbation Bounds for Eigenvalues and Determinants of Matrices. A Survey

Axioms, 2021

The paper is a survey of the recent results of the author on the perturbations of matrices. A part of the results presented in the paper is new. In particular, we suggest a bound for the difference of the determinants of two matrices which refines the well-known Bhatia inequality. We also derive new estimates for the spectral variation of a perturbed matrix with respect to a given one, as well as estimates for the Hausdorff and matching distances between the spectra of two matrices. These estimates are formulated in the terms of the entries of matrices and via so called departure from normality. In appropriate situations they improve the well-known results. We also suggest a bound for the angular sectors containing the spectra of matrices. In addition, we suggest a new bound for the similarity condition numbers of diagonalizable matrices. The paper also contains a generalization of the famous Kahan inequality on perturbations of Hermitian matrices by non-normal matrices. Finally, ta...

Matrix polynomials with specified eigenvalues

Linear Algebra and its Applications, 2015

This work concerns the distance in 2-norm from a matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane. Perturbations are allowed only on the constant coefficient matrix. Singular value optimization formulas are derived for these distances facilitating their computation. The singular value optimization problems, when the number of specified eigenvalues is small, can be solved numerically by exploiting the Lipschitzness and piece-wise analyticity of the singular values with respect to the parameters.

On spectral variations under bounded real matrix perturbations

Numerische Mathematik, 1991

In this paper we investigate the set of eigenvalues of a perturbed matrix A + ∆ ∈ R n×n where A is given and ∆ ∈ R n×n , ∆ < ρ is arbitrary. We determine a lower bound for this spectral value set which is exact for normal matrices A with well separated eigenvalues. We also investigate the behaviour of the spectral value set under similarity transformations. The results are then applied to stability radii which measure the distance of a matrix A from the set of matrices having at least one eigenvalue in a given closed instability domain C b ⊂ C.