A contour integral approach to the computation of invariant pairs

We study some aspects of the invariant pair problem for matrix polynomials, as introduced by Betcke and Kressner [3] and by Beyn and Thümmler [6]. Invariant pairs extend the notion of eigenvalue–eigenvector pairs, providing a counterpart of invariant subspaces for the nonlinear case. We compute formulations for the condition numbers and the backward error for invariant pairs and solvents. We then adapt the Sakurai–Sugiura moment method [1] to the computation of invariant pairs, including some classes of problems that have multiple eigenvalues. Numerical refinement via a variant of Newton's method is also studied. Furthermore, we investigate the relation between the matrix solvent problem and the triangularization of matrix polynomials

On the stabilization of bilinear systems via constant feedback

We study the problem of stabilization of a bilinear system via a constant feedback. The question reduces to an eigenvalue problem on the pencil A+α0B of two matrices. Using the idea of simultaneous triangularization of the matrices involved, some easily checkable conditions for the solvability of this question are obtained. Algorithms for checking these conditions are given and illustrated by a few examples

Stability Of Matrix Polynomials In One And Several Variables

The paper presents methods of eigenvalue localisation of regular matrix polynomials, in particular, stability of matrix polynomials is investigated. For this aim a stronger notion of hyperstability is introduced and widely discussed. Matrix versions of the Gauss-Lucas theorem and Sz\'asz inequality are shown. Further, tools for investigating (hyper)stability by multivariate complex analysis methods are provided. Several second- and third-order matrix polynomials with particular semi-definiteness assumptions on coefficients are shown to be stable.Comment: 19 page

On a class of matrix pencils and ℓ-ifications equivalent to a given matrix polynomial

A new class of linearizations and $ell$-ifications for m×m matrix polynomials $P(x)$ of degree $n$ is proposed. The $ell$-ifications in this class have the form $A(x)=D(x)+(eotimes I_m)W(x)$ where $D$ is a block diagonal matrix polynomial with blocks $B_i(x)$ of size $m$, $W$ is an $m imes qm$ matrix polynomial and $e=(1,…,1)^t in C^q$, for a suitable integer $q$. The blocks $B_i(x)$ can be chosen a priori, subjected to some restrictions. Under additional assumptions on the blocks $B_i(x)$ the matrix polynomial $A(x)$ is a strong $ell$-ification, i.e., the reversed polynomial of $A(x)$ defined by A^# (x):=x^{deg ⁡A(x)}A(x^{-1}) is an $ell$-ification of P^# (x). The eigenvectors of the matrix polynomials $P(x)$ and $A(x)$ are related by means of explicit formulas. Some practical examples of $ell$-ifications are provided. A strategy for choosing $B_i(x)$ in such a way that $A(x)$ is a well conditioned linearization of $P(x)$ is proposed. Some numerical experiments that validate the theoretical results are reported