Related papers: On bundle closures of matrix pencils and matrix po…

Bundles of matrix pencils (under strict equivalence) are sets of pencils having the same Kronecker canonical form, up to the eigenvalues (namely, they are an infinite union of orbits under strict equivalence). The notion of bundle for…

We obtain the generic complete eigenstructures of complex Hermitian $n\times n$ matrix pencils with rank at most $r$ (with $r\leq n$). To do this, we prove that the set of such pencils is the union of a finite number of bundle closures,…

We prove that, given two matrix pencils $L$ and $M$, if $M$ belongs to the closure of the orbit of $L$ under strict equivalence, then the dimension of the orbit of $M$ is smaller than or equal to the dimension of the orbit of $L$, and the…

We show that the set of $n \times n$ complex symmetric matrix pencils of rank at most $r$ is the union of the closures of $\lfloor r/2\rfloor +1$ sets of matrix pencils with some, explicitly described, complete eigenstructures. As a…

Matrix polynomials given in an orthogonal basis are considered. Following the ideas of Mackey et al. "Vector spaces of Linearizations for Matrix Polynomials" (2006), the vec- tor spaces, called M1(P), M2(P) and DM(P), of potential…

The set ${\cal P}^{m\times n}_{r,d}$ of $m \times n$ complex matrix polynomials of grade $d$ and (normal) rank at most $r$ in a complex $(d+1)mn$ dimensional space is studied. For $r = 1, \dots , \min \{m, n\}-1$, we show that ${\cal…

We show that the set of $m \times m$ complex skew-symmetric matrix polynomials of odd grade $d$, i.e., of degree at most $d$, and (normal) rank at most $2r$ is the closure of the single set of matrix polynomials with the certain, explicitly…

The standard way of solving the polynomial eigenvalue problem associated with a matrix polynomial is to embed the matrix polynomial into a matrix pencil, transforming the problem into an equivalent generalized eigenvalue problem. Such…

The seminal work by Mackey et al. in 2006 (reference [21] of the article) introduced vector spaces of matrix pencils, with the property that almost all the pencils in the spaces are strong linearizations of a given square regular matrix…

Given a possibly singular matrix polynomial $P(z)$, we study how the eigenvalues, eigenvectors, root polynomials, minimal indices, and minimal bases of the pencils in the vector space $\mathbb{DL}(P)$ introduced in Mackey, Mackey, Mehl, and…

Let $k$ be a field and $n,a,b$ natural numbers. A matrix pencil $P$ is given by $n$ matrices of the same size with coefficients in $k$, say by $(b\times a)$-matrices, or, equivalently, by $n$ linear transformations $\alpha_i\:k^a \to k^b$…

We study the possible dimension vectors of indecomposable parabolic bundles on the projective line, and use our answer to solve the problem of characterizing those collections of conjugacy classes of n by n matrices for which one can find…

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 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…

We revisit the landmark paper [D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, SIAM J. Matrix Anal. Appl., 28 (2006), pp.~971--1004] and, by viewing matrices as coefficients for bivariate polynomials, we provide concise proofs for key…

The paper deals with root problems for two classes of univariate polynomials both of geometric origin. The first class discussed, the class of Steiner polynomial, consists of polynomials, each associated with a compact convex set V in R^n.…

We obtain the generic real Jordan canonical forms for $n\times n$ matrices with real entries. More precisely, we prove that the set of $n\times n$ real matrices is the union of the closures of $\lfloor n/2\rfloor+1$ sets, which are called…

Let $K$ be a field, $R=K[x, y]$ the polynomial ring and $\mathcal{M}(K)$ the set of all pairs of square matrices of the same size over $K.$ Pairs $P_1=(A_1,B_1)$ and $P_2=(A_2,B_2)$ from $\mathcal{M}(K)$ are called similar if…

The dimensions of sets of matrices of various types, with specified eigenvalue multiplicities, are determined. The dimensions of the sets of matrices with given Jordan form and with given singular value multiplicities are also found. Each…

In this work we approach the problem of determining which (compact) semialgebraic subsets of ${\mathbb R}^n$ are images under polynomial maps $f:{\mathbb R}^m\to{\mathbb R}^n$ of the closed unit ball $\overline{{\mathcal B}}_m$ centered at…