On a linear autonomous descriptor equation with discrete time. II. Canonical representation and structural properties

We consider a linear homogeneous autonomous descriptor equation with discrete time B0g(k+1)+∑mi=1Big(k+1−i)=0,k=m,m+1,…, with rectangular (in general case) matrices Bi. Such an equation arises in the study of the most important control problems for systems with many commensurate delays in control: the 0-controllability problem, the synthesis problem of the feedback-type regulator, which provides calming to the solution of the original system, the modal controllability problem (controllability to the coefficients of characteristic quasipolynomial), the spectral reduction problem and the synthesis problem observers for dual surveillance system. The main method of the presented study is based on replacing the original equation with an equivalent equation in the “expanded” state space, which allows one to match the new equation of the beam of matrices. This made it possible to study a number of structural properties of the original equation by using the canonical form of the beam of matrices, and express the results in terms of minimal indices and elementary divisors. In the article, a criterion is obtained for the existence of a nontrivial admissible initial condition for the original equation, the verification of which is based on the calculation of the minimum indices and elementary divisors of the beam of matrices. The following problem was studied: it is required to construct a solution to the original equation in the form g(k+1)=Tψ(k+1), k=1,2…, where T is some matrix, the sequence of vectors ψ(k+1), k=1,2,…, satisfies the equation ψ(k+1)=Sψ(k), k=1,2,…, and the square matrix S has a predetermined spectrum (or part of the spectrum). The results obtained make it possible to construct solutions of the initial descriptor equation with predetermined asymptotic properties, for example, uniformly asymptotically stable.

Nonnegative Inverse Elementary Divisors Problem for Lists with Nonnegative Real Parts

In this paper, sufficient conditions for the existence and construction of nonnegative matrices with prescribed elementary divisors for a list of complex numbers with nonnegative real part are obtained, and the corresponding nonnegative matrices are constructed. In addition, results of how to perturb complex eigenvalues of a nonnegative matrix while keeping its elementary divisors and its nonnegativity are derived.

Factorizations of Invertible Symmetric Matrices over Polynomial Rings with Involution

By means of the notions of infinite elementary divisors, dual and generalized dual matrix polynomials, we find necessary and sufficient conditions for the existence of factorizations of invertible symmetric matrices over ring of polynomials with involution.

Annihilator-stability and unique generation in C(X)

Using the equivalence of unique generation and cleanness of [Formula: see text], we give affirmative answers to questions raised in [I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949) 464–491] and [D.D. Anderson et al., When are associates unit multiples? Rocky Mountain J. Math. 34 (2004) 811–828] for rings of real-valued continuous functions. In fact, we show that if [Formula: see text] is [Formula: see text] (uniquely generated) then [Formula: see text] is too, and [Formula: see text] is strongly associate if and only if [Formula: see text] is, where [Formula: see text]. We give topological characterizations of [Formula: see text] and [Formula: see text] (annihilator-stable) elements of [Formula: see text] for continuum spaces [Formula: see text] and using this, we observe that the product of two UG elements need not be [Formula: see text]. It is shown that the set of elements in [Formula: see text] which have stable range 1 and the set of [Formula: see text] elements of [Formula: see text] coincide and several examples are given which show that the set of [Formula: see text] elements, the set of [Formula: see text] elements and the set of clean elements of [Formula: see text] can differ. Finally, we characterize spaces [Formula: see text] for which every clean element of [Formula: see text] or every element of [Formula: see text] which has stable range 1 is [Formula: see text].

On the similarity of matrices $AB$ and $BA$ over a field

Let $A$ and $B$ be $n$-by-$n$ matrices over a field. The study of the relationship between the products of matrices $AB$ and $BA$ has a long history. It is well-known that $AB$ and $BA$ have equal characteristic polynomials (and, therefore, eigenvalues, traces, etc.). One beautiful result was obtained by H. Flanders in 1951. He determined the relationship between the elementary divisors of $AB$ and $BA$, which can be seen as a criterion when two matrices $C$ and $D$ can be realized as $C = AB$ and $D = BA$. If one of the matrices ($A$ or $B$) is invertible, then the matrices $AB$ and $BA$ are similar. If both $A$ and $B$ are singular then matrices $AB$ and $BA$ are not always similar. We give conditions under which matrices $AB$ and $BA$ are similar. The rank of matrices plays an important role in this investigation.

Spectra universally realizable by doubly stochastic matrices

A list of complex numbers Λ = { λ1, . . . , λn} is said to be realizable if it is the spectrum of an entrywise nonnegative matrix, and universally realizable if there exists a nonnegative matrix with spectrum Λ for each Jordan canonical form associated with Λ. The problem of characterizing the lists which are universally realizable is called the nonnegative inverse elementary divisors problem (NIEDP). This is a hard problem, which remains unsolved. A complete solution, if any, is still far from the current state of the art in the problem. In particular, in this paper we consider the NIEDP for generalized doubly stochastic matrices, and give new sufficient conditions for the existence and construction of a solution matrix. These conditions improve those given in [ELA 30 (2015) 704-720]

Algebraic generality versus arithmetic generality in the 1874 controversy between C. Jordan and L. Kronecker

This article revisits the 1874 controversy between Camille Jordan and Leopold Kronecker over two theorems, namely Jordan’s canonical forms and Karl Weierstrass’s elementary divisors theorem. In particular, it compares the perspectives of Jordan and Kronecker on generality and how their debate turned into an opposition over the algebraic or arithmetic nature of the ‘theory of forms’. It also examines the ways in which the various actors used the the categories of algebraic generality and arithmetic generality. After providing a background on the Jordan-Kronecker controversy, the article explains Jordan’s canonical reduction and Kronecker’s invariant computations in greater detail. It argues that Jordan and Kronecker aimed to ground the ‘theory of forms’ on new forms of generality, but could not agree on the types of generality and on the treatments of the general they were advocating.