Infinite Matrices Research Articles

Products of infinite upper triangular quadratic matrices

Let F be a field and q(x) a quadratic polynomial in F[x] with q(0)≠0. We denote by T∞(F) the algebra of all infinite upper triangular matrices over the field F. A matrix A∈T∞(F) is called a quadratic matrix with respect to q(x) if q(A)=0. In this paper, we first investigate the subgroup in T∞(F) generated by all quadratic matrices with respect to q(x) and then present some applications.

Banach spaces of sequences arising from infinite matrices

Banach spaces of sequences arising from infinite matrices

Commuting additive maps on upper triangular and strictly upper triangular infinite matrices

Let ${\mathbb F}$ be a field, let $N_{\infty}({\mathbb F})$ be the ring of all ${\mathbb N}\times {\mathbb N}$ strictly upper triangular matrices over ${\mathbb F,}$ and let $T_{\infty}({\mathbb F})$ be the ring of all ${\mathbb N}\times {\mathbb N}$ upper triangular matrices over ${\mathbb F}$. In this paper, we completely characterize additive maps $f:N_{\infty}({\mathbb F})\to T_{\infty}({\mathbb F})$ satisfying $[f(x),x]=0$ for all $x\in N_{\infty}({\mathbb F})$. As applications, we obtain the finite fields versions of the two main results recently obtained by Slowik and Ahmed [Electron. J. Linear Algebra 37:247-255, 2021].

Fixed-point theorem for an infinite Toeplitz matrix and its extension to general infinite matrices

UDC 517.9 In [V. M. Abramov, <em>Bull. Austral. Math. Soc.,</em> <strong>104</strong>, 108–117 (2021)], the fixed-point equation was studied for an infinite nonnegative particular Toeplitz matrix. In the present paper, we provide an alternative proof for the existence of a positive solution in the general case. The presented proof is based on the application of a version of the M. A. Krasnosel'ski fixed-point theorem. The results are then extended to the equations with infinite matrices of a general type.

A Bounded Below, Noncontractible, Acyclic Complex Of Projective Modules

We construct examples of bounded below, noncontractible, acycliccomplexes of finitely generated projective modules over some rings S,as well as bounded above, noncontractible, acyclic complexes ofinjective modules. The rings S are certain rings of infinite matrices with entries inthe rings of commutative polynomials or formal power series ininfinitely many variables. In the world of comodules or contramodules over coalgebras overfields, similar examples exist over the cocommutative symmetriccoalgebra of an infinite-dimensional vector space. A simpler, universal example of a bounded below, noncontractible,acyclic complex of free modules with one generator, communicated tothe author by Canonaco, is included at the end of the paper.

A note on a generalized Jordan form of an infinite upper triangular matrix

In this paper, several equivalent conditions for the existence of a generalized Jordan form for matrices of locally nilpotent linear operators acting on an infinite countable dimensional vector space are presented, and an example is given showing that these conditions are not always fulfilled, which means the existence of infinite upper triangular matrices that are not similar to any generalized infinite Jordan matrix. Besides, it is shown that these conditions are not applicable in the case of an arbitrary dimensional of a vector space.

The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces

Abstract We calculate the essential norm of bounded diagonal infinite matrices acting on Köthe sequence spaces. As a consequence of our result, we obtain a recent criteria for the compactness of multiplication operator acting on Köthe sequence spaces.

A note on Oishi’s lower bound for the smallest singular value of linearized Galerkin equations

Recently Oishi published a paper allowing lower bounds for the minimum singular value of coefficient matrices of linearized Galerkin equations, which in turn arise in the computation of periodic solutions of nonlinear delay differential equations with some smooth nonlinearity. The coefficient matrix of linearized Galerkin equations may be large, so the computation of a valid lower bound of the smallest singular value may be costly. Oishi’s method is based on the inverse of a small upper left principal submatrix, and subsequent computations use a Schur complement with small computational cost. In this note some assumptions are removed and the bounds improved. Furthermore a technique is derived to reduce the total computationally cost significantly allowing to treat infinite dimensional matrices.

A note on products of idempotents in the ring of upper triangular infinite matrices

It is known that every singular matrix of a finite degree defined over a field can be written as a product of a finite number of idempotent matrices. Insipired by this result we investigate the same problem for N × N matrices. It turns out that in this case the analogous theorem does not hold.

Characteristic curves and the exponentiation in the Riordan Lie group: A connection through examples

We point out how to use the classical characteristic method, that is used to solve quasilinear PDE's, to obtain the matrix exponential of some lower triangle infinite matrices. We use the Lie Fréchet structure of the Riordan group described in [4]. After that, we describe some linear dynamical systems in K[[x]] with a concrete involution being a symmetry or a time-reversal symmetry for them. We take this opportunity to assign some dynamical properties to Pascal's triangle.

Opinion Polarization Over Signed Social Networks With Quasi-Structural Balance

In social networks with mixed trust and distrust relationships, polarization is a common class of opinion evolutionary phenomena. Existing studies have generally believed that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">structural balance</i> on networks is a crucial condition for the emergence of opinion polarization under DeGroot-style interactions. Unlike existing studies, this paper aims to establish a new updating model based on DeGroot-style interactions, thereby relaxing the network structure requirement of achieving opinion polarization to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">quasi structural balance</i> , which is a more general concept and can include structural balance as a special case. With the assistance of the method of products convergence of infinite super-stochastic matrices, a comprehensive analysis of the model is carried out, and algebraic conditions on the degrees of trust and distrust among agents are established, so as to ensure the realization of opinion polarization on quasi structurally balanced networks. At last, numerical simulations are provided to verify the theoretical results.

Properties of Infinite Matrices and Sequence Spaces

Some general theorems on the product of matrices and their applications to infinite series have been obtained. All theorems include, as special cases, a set of well-known results. Several specific new results can also be deduced from it.

Associative algebras and the representation theory of grading-restricted vertex algebras

We introduce an associative algebra [Formula: see text] using infinite matrices with entries in a grading-restricted vertex algebra [Formula: see text] such that the associated graded space [Formula: see text] of a filtration of a lower-bounded generalized [Formula: see text]-module [Formula: see text] is an [Formula: see text]-module satisfying additional properties (called a nondegenerate graded [Formula: see text]-module). We prove that a lower-bounded generalized [Formula: see text]-module [Formula: see text] is irreducible or completely reducible if and only if the nondegenerate graded [Formula: see text]-module [Formula: see text] is irreducible or completely reducible, respectively. We also prove that the set of equivalence classes of the lower-bounded generalized [Formula: see text]-modules is in bijection with the set of the equivalence classes of nondegenerate graded [Formula: see text]-modules. For [Formula: see text], there is a subalgebra [Formula: see text] of [Formula: see text] such that the subspace [Formula: see text] of [Formula: see text] is an [Formula: see text]-module satisfying additional properties (called a nondegenerate graded [Formula: see text]-module). We prove that [Formula: see text] are finite-dimensional when [Formula: see text] is of positive energy (CFT type) and [Formula: see text]-cofinite. We prove that the set of the equivalence classes of lower-bounded generalized [Formula: see text]-modules is in bijection with the set of the equivalence classes of nondegenerate graded [Formula: see text]-modules. In the case that [Formula: see text] is a Möbius vertex algebra and the differences between the real parts of the lowest weights of the irreducible lower-bounded generalized [Formula: see text]-modules are less than or equal to [Formula: see text], we prove that a lower-bounded generalized [Formula: see text]-module [Formula: see text] of finite length is irreducible or completely reducible if and only if the nondegenerate graded [Formula: see text]-module [Formula: see text] is irreducible or completely reducible, respectively.

Convergent spectral inclusion sets for banded matrices

AbstractWe obtain sequences of inclusion sets for the spectrum, essential spectrum, and pseudospectrum of banded, in general non‐normal, matrices of finite or infinite size. Each inclusion set is the union of the pseudospectra of certain submatrices of a chosen size n. Via the choice of n, one can balance accuracy of approximation against computational cost, and we show, in the case of infinite matrices, convergence as of the respective inclusion set to the corresponding spectral set.

Isomorphisms of groups of periodic infinite matrices

Isomorphisms of groups of periodic infinite matrices

Generalized Jordan forms of matrices over division rings

Let [Formula: see text] be a division ring that is algebraic over its center [Formula: see text]. The aim of this paper is to describe certain classes of triangular matrices over [Formula: see text] that are similar to generalized Jordan forms, namely: (1) the class of finite triangular matrices, and (2) the class of infinite triangular matrices whose all diagonal entries, except possibly finitely many, belong to [Formula: see text].

Free products of cyclic groups in groups of infinite unitriangular matrices

Groups of infinite unitriangular matrices over associative unitary rings are considered. These groups naturally act on infinite dimensional free modules over underlying rings. They are profinite in case underlying rings are finite. Inspired by their connection with groups defined by finite automata the problem to construct faithful representations of free products of groups by banded infinite unitriangular matrices is considered.For arbitrary prime p a sufficient conditions on a finite set of banded infinite unitriangular matrices over unitary associative rings of characteristic p under which they generate the free product of cyclic p-groups is given. The conditions are based on certain properties of the actions on finite dimensional free modules over underlying rings.It is shown that these conditions are satisfied. For arbitrary free product of finite number of cyclic p-groups constructive examples of the sets of infinite unitriangular matrices over unitar associative rings of characteristic p that generate given free product are presented. These infinite matrices are constructed from finite dimensional ones that are nilpotent Jordan blocks.A few open questions concerning properties of presented examples and other types of faithful representations are formulated.

On Some Sequence Spaces via q-Pascal Matrix and Its Geometric Properties

We develop some new sequence spaces 𝓁p(P(q)) and 𝓁∞(P(q)) by using q-Pascal matrix P(q). We discuss some topological properties of the newly defined spaces, obtain the Schauder basis for the space 𝓁p(P(q)) and determine the Alpha-(α-), Beta-(β-) and Gamma-(γ-) duals of the newly defined spaces. We characterize a certain class (𝓁p(P(q)),X) of infinite matrices, where X∈{𝓁∞,c,c0}. Furthermore, utilizing the proposed results, we characterize certain other classes of infinite matrices. We also examine some geometric properties, like the approximation property, Dunford–Pettis property, Hahn–Banach extension property, and Banach–Saks-type p property of the spaces 𝓁p(P(q)) and 𝓁∞(P(q)).

Flocking Control for Cucker–Smale Model Subject to Denial-of-Service Attacks and Communication Delays

This paper examines the flocking control issue of the Cucker–Smale model in the presence of denial-of-service (DoS) attacks and communication delays. In the setting of DoS attacks, the attacker only obstructs the information communication between agents during the activation phases, while it concentrates on supplying its own energy during the dormancy phases. Furthermore, the communication delays are assumed to be time-varying and heterogeneous. Firstly, a general control input scheme that defends against DoS network attacks and communication delays is constructed. Secondly, on the basis of the presented control input and the properties of graph theory, the flocking control issue is equivalently transformed into a products convergence issue of infinite sub-stochastic matrices. Finally, an algebraic condition is obtained to formulate all the agents that asymptotically achieve the flocking behavior. Moreover, the obtained theoretical results are verified by a numerical example.

Approximation properties of Abel-Poisson integrals on the classes of differentiable functions, defined by means of modulus of continuity

Being the natural apparatus of the periodic functions approximation, the partial Fourier sums are not uniformly convergent over the entire space of the continuous functions. This fact stimulated the search for ways to construct sequences of polynomials that would converge uniformly on the entire space. The matrix method of Fourier series summation is one of the most common methods. Many results on the approximation of the classes of differentiated functions have been obtained for methods generated by triangular infinite matrices. The set of approximating linear methods can be extended by the process of summation of Fourier series, when instead of an infinite triangular matrix one considers the set $\Lambda=\{\lambda_{\delta}(k)\}$ of functions of the natural argument depending on the real parameter $\delta$. The paper deals with the problem of approximation in the uniform metric of $W^{1}H_{\omega}$ classes using one of the classical linear summation methods for Fourier series given by a set of functions of a natural argument, namely, using the Abel-Poisson integral. At the same time, emphasis is placed on the study of the asymptotic behavior of the exact upper limits of the deviations of the Abel-Poisson integrals from the functions of the mentioned class.