Nilpotent Matrices Research Articles

The Zakharov–Shabat Spectral Problem for Complexification and Perturbation of the Korteweg–de Vries Equation

In this paper we consider examples of complex expansion (cKdV) and perturbation (pKdV) of the Korteweg–de Vries equation (KdV) and show that these equations have a representation in the form of the zero-curvature equation. In this case, we use the Lie algebra of 4-dimensional quadratic nilpotent matrices. Moreover, it is shown that the simplest possible matrix representation of this algebra leads to the possibility of constructing a countable number of conservation laws for these equations.

Automatic selfadjoint-ideal semigroups for finite matrices

The notion of automatic selfadjointness of all ideals in a multiplicative semigroup of the bounded linear operators on a separable Hilbert space B(H) arose in a 2015 discussion with Heydar Radjavi who pointed out that B(H) and the finite rank operators F(H) possessed this unitary invariant property which category we named SI semigroups (for automatic selfadjoint ideal semigroups). Equivalent to the SI property is the solvability, for each A in the semigroup, of the bilinear operator equation A⁎=XAY which we believe is a new connection relating the semigroup theory with the theory of operator equations.We found in our earlier works in the subject that even at the basic level of singly generated semigroups, the investigation of SI semigroups led to interesting algebraic and analytic phenomena when generated by rank one operators, normal operators, partial and power partial isometries, subnormal-hyponormal-essentially normal operators, and weighted shift operators; and generated by commuting families of normal operators.In this paper, we focus on a separate Mn(C) treatment for singly generated SI semigroups that requires studying the solvability of the bilinear matrix equation A⁎=XAY in a multiplicative semigroup of finite matrices. This separate focus is needed because the techniques employed in our earlier works we could not adapt to finite matrices. In this paper we find that for certain classes of generators, being a partial isometry is equivalent to generating an SI semigroup. Such classes are: degree 2 nilpotent matrices, weighted shifts, and non-normal Jordan matrices. For the key tools used to establish these equivalences, we developed a number of necessary conditions for singly generated semigroups to be SI for the very general classes: nonselfadjoint matrices, nonzero nilpotent matrices, nonselfadjoint invertible matrices, and Jordan blocks. We also show, for a nonselfadjoint matrix generator in an SI semigroup, the matrix being a partial isometry is equivalent to having norm one. And as an aside, we also prove necessary generator conditions for the SI property when generated by matrices with nonnegative entries.

Decompositions of endomorphisms into a sum of roots of the unity and nilpotent endomorphisms of fixed nilpotence

For n≥2 and fixed k≥1, we study when an endomorphism f of Fn, where F is an arbitrary field, can be decomposed as t+m where t is a root of the unity endomorphism and m is a nilpotent endomorphism with mk=0. For fields of prime characteristic, we show that this decomposition holds as soon as the characteristic polynomial of f is algebraic over its base field and the rank of f is at least nk, and we present several examples that show that the decomposition does not hold in general. Furthermore, we completely solve this decomposition problem for k=2 and nilpotent endomorphisms over arbitrary fields (even over division rings). This somewhat continues our recent publications in Linear Multilinear Algebra (2022) and Int. J. Algebra Comput. (2022) as well as it strengthens results due to Cǎlugǎreanu-Lam in J. Algebra Appl. (2016).

Unipotent diagonalization of matrices

An element $u$ of a ring $R$ is called \textsl{unipotent} if $u-1$ is nilpotent. Two elements $a,b\in R$ are called \textsl{unipotent equivalent} if there exist unipotents $p,q\in R$ such that $b=q^{-1}ap$. Two square matrices $A,B$ are called \textsl{strongly unipotent equivalent} if there are unipotent triangular matrices $P,Q$ with $B=Q^{-1}AP$. In this paper, over commutative reduced rings, we characterize the matrices which are strongly unipotent equivalent to diagonal matrices. For $2\times 2$ matrices over B\'{e}zout domains, we characterize the nilpotent matrices unipotent equivalent to some multiples of $E_{12}$ and the nontrivial idempotents unipotent equivalent to $E_{11}$.

Jordan recoverability of some subcategories of modules over gentle algebras

Gentle algebras form a class of finite-dimensional algebras introduced by I. Assem and A. Skowroński in the 1980s. Modules over such an algebra can be described by string and band combinatorics in the associated gentle quiver from the work of M.C.R. Butler and C.M. Ringel. Any module can be naturally associated to a quiver representation. A nilpotent endomorphism of a quiver representation induces linear transformations over vector spaces at each vertex. Generically among all nilpotent endomorphisms, a well-defined Jordan form exists for these representations. We focus on subcategories additively generated by all the indecomposable representations of a gentle quiver, including a fixed vertex in their support. We show a characterization of the vertices such that the objects of this subcategory are determined up to isomorphism by their generic Jordan form.

Locally finite cycles of linear mappings in countable dimension

Let n be a positive integer. An n-cycle of linear mappings is an n-tuple (u1,…,un) of linear maps u1∈Hom(U1,U2),u2∈Hom(U2,U3),…,un∈Hom(Un,U1), where U1,…,Un are vector spaces over a field. We classify such cycles, up to equivalence, when the spaces U1,…,Un have countable dimension and the composite un∘un−1∘⋯∘u1 is locally finite.When n=1, this problem amounts to classifying the reduced locally nilpotent endomorphisms of a countable-dimensional vector space up to similarity, and the known solution involves the so-called Kaplansky invariants of u. Here, we extend Kaplansky's results to cycles of arbitrary length. As an application, we prove that if un∘⋯∘u1 is locally nilpotent and the Ui spaces have countable dimension, then there are bases B1,…,Bn of U1,…,Un, respectively, such that, for every i∈〚1,n〛, ui maps every vector of Bi either to a vector of Bi+1 or to the zero vector of Ui+1 (where we convene that Un+1=U1 and Bn+1=B1).

Globally hypoelliptic triangularizable systems of periodic pseudo‐differential operators

AbstractThis paper presents an investigation on the global hypoellipticity problem for a class of systems of pseudo‐differential operators on the torus. The approach consists in establishing conditions on the matrix symbol of the system such that it can be transformed into a suitable triangular form involving a nilpotent upper triangular matrix. Hence, the global hypoellipticity is studied by analyzing the behavior of the eigenvalues and their averages.

Spectral properties of solutions of the Yang-Baxter-like matrix equation

We analyse the spectral properties of solutions of the Yang-Baxter-like matrix equation. We explore the solution set when A is nonsingular, give partial results for nilpotent matrices, and construct elementary solutions to the problem.

Direct Deadbeat Control of a Buck PWM Converter

The paper explains the shortcomings of the well-known modal control strategy, which prevent its practical application to achieve the highest possible dynamic performance of switching semiconductor converters. A method for eliminating these shortcomings is suggested as applied to a self-contained modal control version with specifying zero roots of the characteristic polynomial and obtaining a nilpotent system matrix. A technique for synthesizing direct deadbeat control with a nilpotent system matrix is described in detail to implement a finite transient that terminates within a number of cycles not exceeding the order of the difference equation with which a discrete system is described. The control object is a well-known circuit of a DC-DC buck converter with an inductive-capacitive filter and a proportional input-output characteristic provided by an integral link in the feedback loop. The efficiency of the proposed deadbeat controller version is confirmed through mathematically simulating the operation of a PWM converter controlling the load current. The dynamic operation modes are studied for the case of a step change in the load resistance and input voltage relative to their nominal values for a control system with I-, PI- and deadbeat controllers. A comparison of the obtained results shows that the transients in the system equipped with a deadbeat controller terminate most quickly: within three cycles without an overshoot. This corresponds to the maximum convergence rate to the equilibrium state for a third-order system.

Polynomial Liénard systems with a nilpotent global center

A center for a differential system dot{textbf{x}}=f(textbf{x}) in {mathbb {R}}^2 is a singular point p having a neighborhood U such that Usetminus {p} is filled with periodic orbits. A global center is a center p such that {mathbb {R}}^2setminus {p} is filled with periodic orbits. There are three kinds of centers, the centers p such that the Jacobian matrix Df(p) has purely imaginary eigenvalues, the nilpotent centers p such that Df(p) is a nilpotent matrix, and the degenerate centers p such that the matrix Df(p) is the zero matrix. For the first class of centers there are several works studying when such centers are global. As far as we know there are no works for studying the nilpotent global centers. One of the most studied classes of differential systems in {mathbb {R}}^2 are the polynomial Liénard differential systems. The objective of this paper is to study the nilpotent global centers of the polynomial Liénard differential systems.

Mutually annihilating matrices, and a Cohen–Lenstra series for the nodal singularity

We give a generating function for the number of pairs of n×n matrices (A,B) over a finite field that are mutually annihilating, namely, AB=BA=0. This generating function can be viewed as a singular analogue of a series considered by Cohen and Lenstra. We show that this generating function has a factorization that allows it to be meromorphically extended to the entire complex plane. We also use it to count pairs of mutually annihilating nilpotent matrices. This work is essentially a study of the motivic aspects about the variety of modules over C[u,v]/(uv) as well as the moduli stack of coherent sheaves over an algebraic curve with nodal singularities.

On allowable properties and spectrally arbitrary sign pattern matrices

In this paper, we give a geometric construction for different allowable properties for sign pattern matrices. In Section 2, we give a construction for detecting a sign pattern matrix to be potentially nilpotent and also compute the nilpotent matrix realization for a given sign pattern matrix if it exists. In Section 3, we develop a geometric construction for potential stability. In Section 4, we establish a necessary and sufficient condition for a sign pattern matrix S to be spectrally arbitrary. For a given sign pattern matrix of order n, we prove that there exists a surface of dimension at most m for m≤n such that for every vector a1,a2,⋯,an on the same surface, there exists a matrix A∈Q(S), a qualitative class of S whose characteristic polynomial is xn−a1xn−1+⋯+(−1)nan.

On the normalizer of the reflexive cover of a unital algebra of linear transformations

Given a unital algebra A of linear transformations on a finite-dimensional complex vector space V, in this paper we study the set Col(A) consisting of those invertible linear transformations S on V which map every subspace M∈Lat(A) to a subspace SM∈Lat(A). We show that Col(A) is the normalizer of the group of invertible linear transformations in the reflexive cover of A. For the unital algebra (A) which is generated by a linear transformation A, we give the complete description of Col(A). By using the primary decomposition of A, we first reduce the problem of characterizing Col(A) to the problem of characterizing the group Col(N) of a given nilpotent linear transformation N. While Col(N) always contains all invertible linear transformations of the commutant (N)′ of N, it is always contained in the reflexive cover of (N)′. We prove that Col(N) is a proper subgroup of (AlgLat(N)′)−1 if and only if at least two Jordan blocks in the Jordan decomposition of N are of dimension 2 or more. We also determine the group Col(J2⊕J2).

Algorithm for controllable and nilpotent intuitionistic fuzzy matrices

Algorithm for controllable and nilpotent intuitionistic fuzzy matrices

Diagonalizable Matrices as a Result of Rank-One Perturbations of Nilpotent Matrices

Diagonalizable Matrices as a Result of Rank-One Perturbations of Nilpotent Matrices

Spiral waves of divergence in the Barkley model of nilpotent matrices

A new type of a spiral wave is presented in this paper. It is shown that spiral waves of divergence can be generated by the discrete Barkley model when all scalar nodal variables are replaced by two-dimensional nilpotent matrices of variables. However, spiral waves of divergence do not exist if scalar nodal variables of the Barkley model are replaced by idempotent matrices instead. Computational experiments demonstrate that spiral waves diverge along the centerline of the rotating bands, starting from the tip of the spiral wave – but the numerical values of the field stay bounded around zero in the regions between the rotating bands. Spiral waves of divergence are classified into five different classes according to their transient behavior. The formation of transient anti-phase clusters and Wada boundaries of five different types of spiral waves in the parameter plane of the Barkley model are examined in detail. Potential applications of spiral waves of divergence in hiding secret digital images are also discussed.

Cohen Lenstra partitions and mutually annihilating matrices over a finite field

Motivated by questions in algebraic geometry, Yifeng Huang recently derived generating functions for counting mutually annihilating matrices and mutually annihilating nilpotent matrices over a finite field. We give a different derivation of his results using statistical properties of random partitions chosen from the Cohen-Lenstra measure.

Rational solutions of multi‐component nonlinear Schrödinger equation and complex modified KdV equation

In this paper, the crucial condition to achieve rational solutions of the multi‐component nonlinear Schrödinger equation is proposed by introducing two nilpotent Lax matrices. Taking the series multisections of the vector eigenfunction as a set of fundamental eigenfunctions, an explicit formula of the nth‐order rational solution is obtained by the degenerate Darboux transformation, which is used to generate some new patterns of rogue waves. A conjecture about the degree of the nth‐order rogue waves is summarized. This conjecture also holds for rogue waves of the multi‐component complex modified Korteweg–de Vries equation. Finally, the semi‐rational solutions of the Manakov system are discussed.

Decompositions of matrices into potent and square-zero matrices

In order to find a suitable expression of an arbitrary square matrix over an arbitrary finite commutative ring, we prove that every such matrix is always representable as a sum of a potent matrix and a nilpotent matrix of order at most two when the Jacobson radical of the ring has zero-square. This somewhat extends results of ours in Linear Multilinear Algebra (2022) established for matrices considered on arbitrary fields. Our main theorem also improves on recent results due to Abyzov et al. in Mat. Zametki (2017), Šter in Linear Algebra Appl. (2018) and Shitov in Indag. Math. (2019).

Zero-sum triangles for involutory, idempotent, nilpotent and unipotent matrices

In some matrix formations, factorizations and transformations, we need special matrices with some properties and we wish that such matrices should be easily and simply generated and of integers. In this paper, we propose a zero-sum rule for the recurrence relations to construct integer triangles as triangular matrices with involutory, idempotent, nilpotent and unipotent properties, especially nilpotent and unipotent matrices of index 2. With the zero-sum rule we also give the conditions for the special matrices and the generic methods for the generation of those special matrices. Some of the generated integer triangles have been found by other methods, but most of them are newly discovered, and many combinatorial identities can be found with them. The results may also interest the economists for trading analysis and simulation.