相关论文: Generic eigenstructures of Hermitian pencils

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…

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…

Bundles of matrix polynomials are sets of matrix polynomials with the same size and grade and the same eigenstructure up to the specific values of the eigenvalues. It is known that the closure of the bundle of a pencil $L$ (namely, a matrix…

In Parts I and II of this series of papers, three new methods for the computation of eigenvalues of singular pencils were developed: rank-completing perturbations, rank-projections, and augmentation. It was observed that a straightforward…

We determine the generic complete eigenstructures for $n \times n$ complex symmetric matrix polynomials of odd grade $d$ and rank at most $r$. More precisely, we show that the set of $n \times n$ complex symmetric matrix polynomials of odd…

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…

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 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…

Pseudo-hermitian matrices are matrices hermitian with respect to an indefinite metric. They can be thought of as the truncation of pseudo-hermitian operators, defined over some Krein space, together with the associated metric, to a finite…

We analyze when an arbitrary matrix pencil is equivalent to a dissipative Hamiltonian pencil and show that this heavily restricts the spectral properties. In order to relax the spectral properties, we introduce matrix pencils with…

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 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…

Hermitian linear matrix pencils are ubiquitous in control theory, operator systems, semidefinite optimization, and real algebraic geometry. This survey reviews the fundamental features of the matricial solution set of a linear matrix…

We study singular Hermitian metrics on vector bundles. There are two main results in this paper. The first one is on the coherence of the higher rank analogue of multiplier ideals for singular Hermitian metrics defined by global sections.…

We give a complete characterization of invariant integrable complex structures on principal bundles defined over hermitian symmetric spaces, using the Jordan algebraic approach for the curvature computations. In view of possible…

When computing the eigenstructure of matrix pencils associated with the passivity analysis of perturbed port-Hamiltonian descriptor system using a structured generalized eigenvalue method, one should make sure that the computed spectrum…

In this manuscript, a generalized inverse eigenvalue problem is considered that involves a linear pencil $(z\mathcal{J}_{[0,n]}-\mathcal{H}_{[0,n]})$ of matrices arising in the theory of rational interpolation and biorthogonal rational…

We prove using a novel random matrix model that all right-angled Artin groups have a sequence of finite dimensional unitary representations that strongly converge to the regular representation. We deduce that this result applies also to:…

We give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models, assuming that the potential that determines the model is globally $C^{2}$ and locally $C^{3}$ function (see Theorem \ref{t:U.t1}). The…

This paper presents a fast, randomized divide-and-conquer algorithm for the definite generalized eigenvalue problem, which corresponds to pencils $(A,B)$ in which $A$ and $B$ are Hermitian and the Crawford number $\gamma(A,B) =…