Eigenvalue Placement for Regular Matrix Pencils with Rank One Perturbations

Abstract

A regular matrix pencil sE − A and its rank one perturbations are considered. We determine the sets in C ∪ {∞} which are the eigenvalues of the perturbed pencil. We show that the largest Jordan chains at each eigenvalue of sE − A may disappear and the sum of the length of all destroyed Jordan chains is the number of eigenvalues (counted with multiplicities) which can be placed arbitrarily in C ∪ {∞}. We prove sharp upper and lower bounds of the change of the algebraic and geometric multiplicity of an eigenvalue under rank one perturbations. Finally we apply our results to a pole placement problem for a single-input differential algebraic equation with feedback.Keywords: regular matrix pencils, rank one perturbations, spectral perturbation theory MSC 2010: 15A22, 15A18, 47A55 S(sE − A)T = s I r 0 0 N − J 0 0 I n−r , r ∈ {0, 1, . . . , n}

Weierstrass Structure and Eigenvalue Placement of Regular Matrix Pencils under Low Rank Perturbations

“…In other cases, the matrix P is an arbitrary perturbation belonging to the whole set of matrices of rank less than or equal to r. In this paper we follow the second approach. Some results related to the problem from this point of view can be found in [6,9,11,13,19,20,21,22]. We next analyze most of them.…”

Fixed rank perturbations of regular matrix pencils

“…For regular pencils the problem has been studied for r = 1 in [15]. For arbitrary perturbations of bounded rank the problem has been solved in [2], and for perturbations of fixed rank in [1].…”

Generic skew-symmetric matrix polynomials with fixed rank and fixed odd grade

“…See for instance the references [2,3,4,8,11,17,43] for different low rank perturbation problems related to matrix pencils, and the particular study in [10] on certain ✩ Preprint Report UMINF 17.07, Department of Computing Science, Umeå University Email addresses: andrii@cs.umu.se (Andrii Dmytryshyn), dopico@math.uc3m.es (Froilán M. Dopico) low rank perturbations of matrix polynomials. Moreover, low rank perturbations of matrix pencils have been applied recently to some classical problems as the eigenvalue placement problem [30] or the estimation of the distance of a regular pencil to the nearest singular pencil [42]. Finally, from a more theoretical perspective, the study of the sets of matrix pencils and of matrix polynomials with fixed grade and fixed rank generalizes classical studies [48] on the algebraic structure of the set of n×n singular pencils, i.e., those whose rank is at most n − 1.…”