Related papers: Rank-one perturbations of matrix pencils
The change of the Kronecker structure of a matrix pencil perturbed by another pencil of rank one has been characterized in terms of the homogeneous invariant factors and the chains of column and row minimal indices of the initial and the…
A characterization of the structure of a regular matrix pencil obtained by a bounded rank perturbation of another regular matrix pencil has been recently obtained. The result generalizes the solution for the bounded rank perturbation…
The complete characterization of the Kronecker structure of a matrix pencil perturbed by another pencil of rank one is known, and it is stated in terms of very involved conditions. This paper is devoted to, loosing accuracy, better…
The relationship between linear relations and matrix pencils is investigated. Given a linear relation, we introduce its Weyr characteristic. If the linear relation is the range (or the kernel) representation of a given matrix pencil, we…
We solve the problem of determining the Weierstrass structure of a regular matrix pencil obtained by a low rank perturbation of another regular matrix pencil. We apply the result to find necessary and sufficient conditions for the existence…
Under the action of the general linear group with tensor structure, the ranks of matrices $A$ and $B$ forming an $m \times n$ pencil $A + \lambda B$ can change, but in a restricted manner. Specifically, with every pencil one can associate a…
A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory results for linearizations need to be related back to matrix polynomials. In this paper we…
The generic change of the Weierstrass Canonical Form of regular complex structured matrix pencils under generic structure-preserving additive low-rank perturbations is studied. Several different symmetry structures are considered and it is…
Translated into the language of representations of quivers, a challenge in matrix pencil theory is to find sufficient and necessary conditions for a Kronecker representation to be a subfactor of another Kronecker representation in terms of…
Our purpose is to give new proofs of several known results about perturbations of matrix pencils. Andrzej Pokrzywa (1986) described the closure of orbit of a Kronecker canonical pencil $A-\lambda B$ in terms of inequalities with pencil…
A regular matrix pencil sE-A and its rank one perturbations are considered. We determine the sets in the extended complex plane which are the eigenvalues of the perturbed pencil. We show that the largest Jordan chains at each eigenvalue of…
We elaborate on the deviation of the Jordan structures of two linear relations that are finite-dimensional perturbations of each other. We compare their number of Jordan chains of length at least $n$. In the operator case, it was recently…
The stated paper is dedicated to one of the inverse problems of spectral theory. It is necessary to define matrix (constant) coefficients of some quadratic pencil, if the eigenvalues of this pencil are known. Furthermore, it is known that…
We give a method for constructing a regularizing decomposition of a matrix pencil, which is formulated in terms of the linear mappings. We prove that two pencils are topologically equivalent if and only if their regularizing decompositions…
Matrix pencils, or pairs of matrices, may be used in a variety of applications. In particular, a pair of matrices (E,A) may be interpreted as the differential equation E x' + A x = 0. Such an equation is invariant by changes of variables,…
We study tensor network varieties associated with the triangular graph, with a focus on the case where one of the physical dimensions is 2. This allows us to interpret the tensors as pencils of matrices. We provide a complete…
We study linear preserver problems on the linear space of $n\times n$ Toeplitz matrices over the real field or the complex field. In particular, characterizations are given for linear preservers of rank one matrices and linear preservers of…
The set of mxn singular matrix pencils with normal rank at most r is an algebraic set with r+1 irreducible components. These components are the closure of the orbits (under strict equivalence) of r+1 matrix pencils which are in Kronecker…
The multiparameter matrix pencil problem (MPP) is a generalization of the one-parameter MPP: given a set of $m\times n$ complex matrices $A_0,\ldots, A_r$, with $m\ge n+r-1$, it is required to find all complex scalars…
The results of [I. Ojeda, Amer. Math. Monthly, 122, pp 60--64] provides a characterization of Kronecker square roots of matrices in terms of the symmetry and rank of the block vec matrix (rearrangement matrix). In this short note we…