Developed in 1999 by Akemann, Anderson, and Weaver, the spectral scale of an $n\times n$ matrix $A$, is a convex, compact subset of $\mathbb{R}^3$ that reveals important spectral information about $A$ \cite{AAW}. In this paper we present new information found in the spectral scale of a matrix. Given a matrix $A=A_1 + iA_2$ with $A_1$ and $A_2$ self-adjoint and $A_2\neq 0,$ we show that faces in the boundary of the spectral scale of $A$ that are parallel to the x-axis describe elements of $σ(A_1,A_2)\bigcap\mathbb{R},$ the real elements of the spectrum of the linear pencil $P(λ)=A_1 + λA_2.$
We introduce concepts of essential numerical range for the linear operator pencil $λ\mapsto A-λB$. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor even connected. The new concepts allow us to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. Moreover, by transforming the operator eigenvalue problem $Tx=λx$ into the pencil problem $BTx=λBx$ for suitable choices of $B$, we can obtain non-convex spectral enclosures for $T$ and, in the study of truncati…
We consider a second order functional-differential pencil with two constant delays of the argument and study the inverse problem of recovering its coefficients from the spectra of two boundary value problems with one common boundary condition. The uniqueness theorem is proved and a constructive procedure for solving this inverse problem along with necessary and sufficient conditions for its solvability is obtained. Moreover, we give a survey on the contemporary state of the inverse spectral theory for operators with delay. The pencil under consideration generalizes Sturm-Liouville-type operato…
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 show that there is a correspondence between this characteristic and the Kronecker canonical form of the pencil. This relationship is exploited to obtain estimations on the invariant characteristics of matrix pencils under rank one perturbations.
The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local solvability and stability. The problem is considered in the general case of complex-valued pencil coefficients and arbitrary eigenvalue multiplicities. Studying local solvability and stability, we take the possible splitting of multiple eigenvalues under a small perturbation of the spectrum into account. Our approach is constructive. It is based on the reduction of…
The relation between the spectra of operator pencils with unbounded coefficients and of associated linear relations is investigated. It turns out that various types of spectrum coincide and the same is true for the Weyr characteristics. This characteristic describes how many independent Jordan chains up to a certain length exist. Furthermore, the change of this characteristic subject to one-dimensional perturbations is investigated.
In the paper, we study an inverse spectral problem for quadratic pencils of the Sturm--Liouville operators with singular coefficients and entire functions in the boundary conditions. We prove that a subspectrum is sufficient for recovering the pencil if this subspectrum generates a complete functional system. As well, we obtain an algorithm solving the inverse problem and alternative conditions on the subspectrum. Finally, these results are applied to studying a partial inverse problem.
We establish spectral enclosures and spectral approximation results for the inhomogeneous lossy Drude-Lorentz system with purely imaginary poles, in a possibly unbounded Lipschitz domain of $\mathbb{R}^3$. Under the assumption that the coefficients $θ_e$, $θ_m$ of the material are asymptotically constant at infinity, we prove that: 1) the essential spectrum can be decomposed as the union of the spectrum of a bounded operator pencil in the form $- \operatorname{div} p(ω) \nabla$ and of a second order $\operatorname{curl} \operatorname{curl}_0 - V_{e,\infty}(ω)$ pencil with constant coefficients…
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 matrix pencils was introduced in the 1990's, following the same notion for matrices under similarity, introduced by Arnold in 1971, and it has been extensively used since then. Despite the amount of literature devoted to describing the topology of bundles of matrix pencils, some relevant questions remain still open in this context. For example, the following two:…
Linearization is a standard method in the computation of eigenvalues and eigenvectors of matrix polynomials. In the last decade a variety of linearization methods have been developed in order to deal with algebraic structures and in order to construct efficient numerical methods. An important source of linearizations for matrix polynomials are the so called Fiedler pencils, which are generalizations of the Frobenius companion form and these linearizations have been extended to regular rational matrix function which is the transfer function of LTI State-space system in [1, 6]. We consider a mul…
We study the backward stability of running a backward stable eigenstructure solver on a pencil $S(λ)$ that is a strong linearization of a rational matrix $R(λ)$ expressed in the form $R(λ)=D(λ)+ C(λI_\ell-A)^{-1}B$, where $D(λ)$ is a polynomial matrix and $C(λI_\ell-A)^{-1}B$ is a minimal state-space realization. We consider the family of block Kronecker linearizations of $R(λ)$, which are highly structured pencils. Backward stable eigenstructure solvers applied to $S(λ)$ will compute the exact eigenstructure of a perturbed pencil $\widehat S(λ):=S(λ)+Δ_S(λ)$ and the special structure of $S(λ)…
