Related papers: Reduction and Normal Forms of Matrix Pencils
Matrix pencils, or pairs of matrices, are used in a variety of applications. By the Kronecker decomposition Theorem, they admit a normal form. This normal form consists of four parts, one part based on the Jordan canonical form, one part…
The reductions of a square complex matrix A to its canonical forms under transformations of similarity, congruence, or *congruence are unstable operations: these canonical forms and reduction transformations depend discontinuously on the…
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…
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…
We analyze different approaches to differential-algebraic equations with attention to the implemented rank conditions of various matrix functions. These conditions are apparently very different and certain rank drops in some matrix…
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…
To understand the solution of a linear, time-invariant differential-algebraic equation, one must analyze a matrix pencil (A,E) with singular E. Even when this pencil is stable (all its finite eigenvalues fall in the left-half plane), the…
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 group $SL(2,\mathbb{C})$ of all complex $2\times 2$ matrices with determinant one is closely related to the group $\boldsymbol{\mathcal{L}}_{+}^\uparrow$ of real $4\times 4$ matrices representing the restricted Lorentz transformations.…
For each square complex matrix, V. I. Arnold constructed a normal form with the minimal number of parameters to which a family of all matrices B that are close enough to this matrix can be reduced by similarity transformations that smoothly…
We introduce a new class of structured matrix polynomials, namely, the class of M_A-structured matrix polynomials, to provide a common framework for many classes of structured matrix polynomials that are important in applications: the…
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…
For each pair of complex symmetric matrices $(A,B)$ we provide a normal form with a minimal number of independent parameters, to which all pairs of complex symmetric matrices $(\widetilde{A},\widetilde{B})$, close to $(A,B)$ can be reduced…
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…
A problem that is frequently encountered in a variety of mathematical contexts, is to find the common invariant subspaces of a single, or set of matrices. A new method is proposed that gives a definitive answer to this problem. The key idea…
The aim of this paper is twofold. First, we introduce a new class of linearizations, based on the generalization of a construction used in polynomial algebra to find the zeros of a system of (scalar) polynomial equations. We show that one…
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…
Compound matrices play an important role in many fields of mathematics and have recently found new applications in systems and control theory. However, the explicit formulas for these compounds are non-trivial and not always easy to use.…
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…
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 $\alpha_i\:k^a \to k^b$…