Elementary Divisors Research Articles

Elementary divisors, Hochster duality, and spectra

Elementary divisors, Hochster duality, and spectra

Parametrizing spectral filters for nonsingular polynomial matrices

Two polynomial matrices are spectrally equivalent if they have the same finite and infinite elementary divisors. Spectral equivalence can be characterized using very special polynomial matrices called spectral filters. The main results in this paper concern the parametrization of spectral filters for any given pair of nonsingular spectrally equivalent polynomial matrices of arbitrary degree d. First we parametrize the set of their spectral filters of degree less than d when the given spectrally equivalent matrices have nonsingular leading coefficients. The parameter space is the subset of invertible matrices of the centralizer of any of their linearizations. When the leading coefficients are singular, we provide a parametrization of the set of spectral filters of degree d−1 and show how to obtain the spectral filters of degree less than d−1. Now the parameter space is the subset of invertible matrices of the centralizer of any linearization of the reversals with respect to a scalar that is not an eigenvalue of the given matrices. Ideas from the Theory of Linear Control Systems are used to introduce left and right realizations of matrix polynomials. These are the building blocks on which the parameterization of spectral filters is constructed. The results are illustrated with examples.

On adequacy of full matrices

This paper deals with the following question:whether a ring of matrices or classes of matrices over an adequate ring or elementary divisor ring inherits the property of adequacy?
 The property to being adequate in matrix rings over adequate and commutative elementary divisor rings is studied.Let us denote by $\mathfrak{A}$ and $\mathfrak{E}$ an adequate and elementary divisor domains, respectively. Also $\mathfrak{A}_2$ and $\mathfrak{E}_2$ denote a rings of $2 \times 2$ matrices over them. We prove that full nonsingular matrices from $\mathfrak{A}_2$ are adequate in $\mathfrak{A}_2$ and full singular matrices from $\mathfrak{E}_2$ are adequate in the set of full matrices in $\mathfrak{E}_2$.

Asymptotic solutions of singularly perturbed linear differential-algebraic equations with periodic coefficients

The paper deals with the problem of constructing asymptotic solutions for singular perturbed linear differential-algebraic equations with periodic coefficients. The case of multiple roots of a characteristic equation is studied. It is assumed that the limit pencil of matrices of the system has one eigenvalue of multiplicity n, which corresponds to two finite elementary divisors and two infinite elementary divisors whose multiplicity is greater than 1.A technique for finding the asymptotic solutions is developed and n formal linearly independent solutions are constructed for the corresponding differential-algebraic system. The developed algorithm for constructing formal solutions of the system is a nontrivial generalization of the corresponding algorithm for constructing asymptotic solutions of a singularly perturbed system of differential equations in normal form, which was used in the case of simple roots of the characteristic equation.The modification of the algorithm is based on the equalization method in a special way the coefficients at powers of a small parameter in algebraic systems of equations, from which the coefficients of the formal expansions of the searched solution are found. Asymptotic estimates for the terms of these expansions with respect to a small parameter are also given.For an inhomogeneous differential-algebraic system of equations with periodic coefficients, existence and uniqueness theorems for a periodic solution satisfying some asymptotic estimate are proved, and an algorithm for constructing the corresponding formal solutions of the system is developed. Both critical and non-critical cases are considered.

Expanders on matrices over a finite chain ring, I

In this work and its sequel, we study the expanding phenomenon of matrices over a finite chain ring of large residue field. A sum-product estimate is proved. It is shown that [Formula: see text] is a moderate expander on [Formula: see text] matrices with exponent [Formula: see text]. These results generalize the main theorems in the recent work [29] of Xie and Ge. The proofs use spectral graph theory and elementary divisor theory.

Equivalent generating vectors of finitely generated modules over commutative rings

Let R be a commutative ring with identity and let M be an R -module which is generated by μ elements but not fewer. We denote by SL n ( R ) the group of the n × n matrices over R with determinant 1. We denote by E n ( R ) the subgroup of SL n ( R ) generated by the matrices which differ from the identity by a single off-diagonal coefficient. Given n ≥ μ and G ∈ { SL n ( R ) , E n ( R ) } , we study the action of G by matrix right-multiplication on V n ( M ) , the set of elements of M n whose components generate M . Assuming that M is finitely presented and that R is an elementary divisor ring or an almost local-global coherent Prüfer ring, we obtain a description of V n ( M ) / G which extends the author's earlier result on finitely generated modules over quasi-Euclidean rings [22] .

Quadratic realizability of palindromic matrix polynomials: the real case

Let be a list consisting of structural data for a matrix polynomial; here is a sublist consisting of powers of irreducible (monic) scalar polynomials over the field , and is a sublist of nonnegative integers. For an arbitrary such , we give easy-to-check necessary and sufficient conditions for to be the list of elementary divisors and minimal indices of some real T-palindromic quadratic matrix polynomial. For a list satisfying these conditions, we show how to explicitly build a real T-palindromic quadratic matrix polynomial having as its structural data; that is, we provide a T-palindromic quadratic realization of over . A significant feature of our construction differentiates it from related work in the literature; the realizations constructed here are direct sums of blocks with low bandwidth, that transparently display the spectral and singular structural data in the original list .

Stable range conditions for abelian and duo rings

The article deals with the following question: when does the classical ring of quotientsof a duo ring exist and idempotents in the classical ring of quotients $Q_{Cl} (R)$ are thereidempotents in $R$? In the article we introduce the concepts of a ring of (von Neumann) regularrange 1, a ring of semihereditary range 1, a ring of regular range 1. We find relationshipsbetween the introduced classes of rings and known ones for abelian and duo rings.We proved that semihereditary local duo ring is a ring of semihereditary range 1. Also it was proved that a regular local Bezout duo ring is a ring of stable range 2. In particular, the following Theorem 1 is proved: For an abelian ring $R$ the following conditions are equivalent:$1.$\ $R$ is a ring of stable range 1; $2.$\ $R$ is a ring of von Neumann regular range 1.
 The paper also introduces the concept of the Gelfand element and a ring of the Gelfand range 1 for the case of a duo ring. Weproved that the Hermite duo ring of the Gelfand range 1 is an elementary divisor ring (Theorem 3).

Linear systems of Diophantine equations

Given free modules $M\subseteq L$ of finite rank $f\geq 1$ over a principal ideal domain $R$, we give a procedure to construct a basis of $L$ from a basis of $M$ assuming the invariant factors or elementary divisors of $L/M$ are known. Given a matrix $A\in M_{m,n}(R)$ of rank $r$, its nullspace $L$ in $R^n$ is a free $R$-module of rank $f=n-r$. We construct a free submodule $M$ of $L$ of rank $f$ naturally associated with $A$ and whose basis is easily computable, we determine the invariant factors of the quotient module $L/M$ and then indicate how to apply the previous procedure to build a basis of $L$ from one of $M$.

On a formula of Thompson and McEnteggert for the adjugate matrix

For an eigenvalue λ0 of a Hermitian matrix A, the formula of Thompson and McEnteggert gives an explicit expression of the adjugate of λ0I−A, Adj(λ0I−A), in terms of eigenvectors of A for λ0 and all its eigenvalues. In this paper Thompson-McEnteggert's formula is generalized to include any matrix with entries in an arbitrary field. In addition, for any nonsingular matrix A, a formula for the elementary divisors of Adj(A) is provided in terms of those of A. Finally, a generalization of the eigenvalue-eigenvector identity and three applications of the Thompson-McEnteggert's formula are presented.

Limits and fluctuations of p-adic random matrix products

We show that singular numbers (also known as elementary divisors, invariant factors or Smith normal forms) of products and corners of random matrices over $${\mathbb {Q}}_p$$ are governed by the Hall–Littlewood polynomials, in a structurally identical manner to the known relations between singular values of complex random matrices and Heckman–Opdam hypergeometric functions. This implies that the singular numbers of a product of corners of Haar-distributed elements of $$\mathrm {GL}_N({\mathbb {Z}}_p)$$ form a discrete-time Markov chain distributed as a Hall–Littlewood process, with the number of matrices in the product playing the role of time. We give an exact sampling algorithm for the Hall–Littlewood processes which arise by relating them to an interacting particle system similar to PushTASEP. By analyzing the asymptotic behavior of this particle system, we show that the singular numbers of such products obey a law of large numbers and their fluctuations converge dynamically to independent Brownian motions. In the limit of large matrix size, we also show that the analogues of the Lyapunov exponents for matrix products have universal limits within this class of $$\mathrm {GL}_N({\mathbb {Z}}_p)$$ corners.

The Left Greatest Common Divisor and the Left Least Common Multiple for All Solutions of the Matrix Equation BX = A Over a Commutative Domain of Elementary Divisors

The Left Greatest Common Divisor and the Left Least Common Multiple for All Solutions of the Matrix Equation BX = A Over a Commutative Domain of Elementary Divisors

The reciprocal character of the conjugation action

For a finite group $G$ we investigate the smallest positive integer $e(G)$ such that the map sending $g\in G$ to $e(G)|G:C_G(g)|$ is a generalized character of $G$. It turns out that $e(G)$ is strongly influenced by local data, but behaves irregularly for non-abelian simple groups. We interpret $e(G)$ as an elementary divisor of a certain non-negative integral matrix related to the character table of $G$. Our methods applied to Brauer characters also answers a recent question of Navarro: The $p$-Brauer character table of $G$ determines $|G|_{p'}$.

Finitely Generated Modules's Uniserial Dimensions Over a Discrete Valuation Domain

We present some methods for calculating the module's uniserial dimension that finitely generated over a DVD in this article. The idea of a module's uniserial dimension over a commutative ring, which defines how far the module deviates from being uniserial, was recently proposed by Nazemian etc. They show that if R is Noetherian commutative ring, which implies that every finitely generated module over R has uniserial dimension. Ghorbani and Nazemians have shown that R is Noetherian (resp. Artinian) ring if only the ring R X R has (resp. finite) valuation dimension. The finitely generated modules over valuation domain are further examined from here. However, since the region remains too broad, further research into the module's uniserial dimensions that finitely generated over a DVD is needed. In the case of a DVD R, a finitely generated module over R can, as is well-known, be divided into a direct sum of torsion and a free module. Therefore, first, we present methods for determining the primary module's uniserial dimension, and then followed by methods for the general finitely generated module. As can be observed, the module's uniserial dimension is a function of the elementary divisors and the rank of the non torsion module item, which is the major finding of this work.

Clear rings and clear elements

An element of a ring $R$ is called clear if it is a sum of a unit-regular element and a unit. An associative ring is clear if each of its elements is clear.In this paper we defined clear rings and extended many results to a wider class. Finally, we proved that a commutative Bezout domain is an elementary divisor ring if and only if every full $2\times 2$ matrix over it is nontrivially clear.

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.

Spectral filters connecting high order systems

Three criteria are given to characterize when two linear dynamical systems have the same spectral structure (same finite and infinite elementary divisors). They are allowed to have different orders or sizes and their leading coefficient may be singular. One of the criteria uses generalized reversal matrix polynomials, while the others rely on the existence of spectral filters. These are matrix polynomials which play a similar role to the change of bases for first order systems. A constructive procedure is presented to obtain spectral filters linking any two systems with the same spectral structure. Connections are made with the second-order systems decoupling problem.

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.

Almost inner derivations of 2-step nilpotent Lie algebras of genus 2

We study almost inner derivations of 2-step nilpotent Lie algebras of genus 2, i.e., having a 2-dimensional commutator ideal, using matrix pencils. In particular we determine all almost inner derivations of such algebras in terms of minimal indices and elementary divisors over an arbitrary algebraically closed field of characteristic not 2 and over the real numbers.

Revisiting the low-rank eigenvalue problem

In this paper, we are interested in the eigenproblem on the large and low-rank matrix S=ABH, where A,B∈ℂn×r are of full column rank and r≪n. To the best of our knowledge, there are no results on the relations between the Jordan decomposition and the Schur decomposition of BHA and those of ABH. Some known results are only on characteristic polynomials, elementary divisors, and Jordan blocks of ABH, and are purely theoretical and are not easy to use for computational purposes. Based on the Jordan decomposition and the Schur decomposition of the small matrix BHA∈ℂr×r, we consider how to derive those of the large matrix AHB∈ℂn×n in this work. The construction methods proposed are not only theoretical but also practical. Numerical experiments show the effectiveness of our theoretical results.