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

A blender is a closed convex cone of real homogeneous polynomials that is also closed under linear changes of variable. Non-trivial blenders only occur in even degree. Examples include the cones of psd forms, sos forms, convex forms and…

In the first part of this paper, the main concern is with smoothness properties of the boundary of the pseudospectrum of a matrix polynomial. In the second part, results are obtained concerning the number of connected components of…

In this paper, we introduce a new family of equations for matrix pencils that may be utilized for the construction of strong linearizations for any square or rectangluar matrix polynomial. We provide a comprehensive characterization of the…

A linear principal minor polynomial or lpm polynomial is a linear combination of principal minors of a symmetric matrix. By restricting to the diagonal, lpm polynomials are in bijection to multiaffine polynomials. We show that this…

The main objective of this talk is to develop a matrix pencil approach for the study of an initial value problem of a class of singular linear matrix differential equations whose coefficients are constant matrices. By using matrix pencil…

We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\Pi_{2^{m}}^*$ represent the set of polynomials computable with…

The matrix pencil (A,B) = {tB-A | t \in C} is considered under the assumptions that A is entrywise nonnegative and B-A is a nonsingular M-matrix. As t varies in [0,1], the Z-matrices tB-A are partitioned into the sets L_s introduced by…

Let A be a k-vector space of dimension a. A subvector space M of End(A) is said to be of rank r if every non-zero f in M has rank r. The problem considered in this paper is to determine l(r;a) the maximal dimension of a rank r subspace of…

This paper solves the two-sided version and provides a counterexample to the general version of the 2003 conjecture by Hadwin and Larson. Consider evaluations of linear matrix pencils $L=T_0+x_1T_1+\cdots+x_mT_m$ on matrix tuples as…

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 characterizing the existence of a polynomial matrix of fixed degree when its eigenstructure (or part of it) and some of its rows (columns) are prescribed. More specifically, we present a solution to the row (column)…

A new class of linearizations and $\ell$-ifications for $m\times m$ matrix polynomials $P(x)$ of degree $n$ is proposed. The $\ell$-ifications in this class have the form $A(x) = D(x) + (e\otimes I_m) W(x)$ where $D$ is a block diagonal…

Strongly $\mathbb{Z}$-graded algebras or principal circle bundles and associated line bundles or invertible bimodules over a class of generalized Weyl algebras $\mathcal{B}(p;q, 0)$ (over a ring of polynomials in one variable) are…

Let $C$ be an algebraic curve defined by a sufficiently generic bivariate Laurent polynomial with given Newton polygon $\Delta$. It is classical that the geometric genus of $C$ equals the number of lattice points in the interior of…

One useful standard method to compute eigenvalues of matrix polynomials ${\bf P}(z) \in \mathbb{C}^{n\times n}[z]$ of degree at most $\ell$ in $z$ (denoted of grade $\ell$, for short) is to first transform ${\bf P}(z)$ to an equivalent…

We present closed form expressions for the ranks of all cohomology groups of holomorphic line bundles on several Calabi-Yau threefolds realised as complete intersections in products of projective spaces. The formulae have been obtained by…

We investigate the minimal singularities of metrics on a big line bundle $L$ over a projective manifold when the stable base locus $Y$ of $L$ is a submanifold of codimension $r\geq 1$. Under some assumptions on the normal bundle and a…

In this work, we characterize matrices of linear forms and constant rank, demonstrating that, under some natural assumptions, they are always associated with a syzygy bundle that fits into a (partially linear) resolution. Furthermore, this…

A number $\lambda \in \mathbb C $ is called an {\it eigenvalue} of the matrix polynomial $P(z)$ if there exists a nonzero vector $x \in \mathbb C^n$ such that $P(\lambda)x = 0$. Note that each finite eigenvalue of $P(z)$ is a zero of the…

The main purpose of this article is to show that the numerical range of a linear pencil $\lambda A + B$ is equal to $\mathbb{C}$ if and only if $0$ belongs to the convex hull of the joint numerical range of $A$ and $B$. We also prove that…