Nilpotent Matrices Research Articles

Nilpotent matrices having a given Jordan type as maximum commuting nilpotent orbit

The Jordan type of a nilpotent matrix is the partition giving the sizes of its Jordan blocks. We study pairs of partitions (P,Q), where Q=Q(P) is the Jordan type of a generic nilpotent matrix A commuting with a nilpotent matrix B of Jordan type P. T. Košir and P. Oblak have shown that Q has parts that differ pairwise by at least two. Such partitions, which are also known as “super distinct” or “Rogers–Ramanujan”, are exactly those that are stable or “self-large” in the sense that Q(Q)=Q.In 2012 P. Oblak formulated a conjecture concerning the cardinality of Q−1(Q) when Q has two parts, and proved some special cases. R. Zhao refined this to posit that the partitions in Q−1(Q) for Q=(u,u−r) with u>r>1 could be arranged in an (r−1)×(u−r) table T(Q) where the entry in the k-th row and ℓ-th column has k+ℓ parts. We prove this Table Theorem, and then generalize the statement to propose a Box Conjecture for the set of partitions Q−1(Q) for an arbitrary partition Q whose parts differ pairwise by at least two.

Commuting Solutions of a Quadratic Matrix Equation for Nilpotent Matrices

We solve the quadratic matrix equation AXA = XAX with a given nilpotent matrix A, to find all commuting solutions. We first provide a key lemma, and consider the special case that A has only one Jordan block to motivate the idea for the general case. Our main result gives the structure of all the commuting solutions of the equation with an arbitrary nilpotent matrix.

When is every linear transformation a sum of an idempotent one and a locally nilpotent one?

For a semisimple module M over a ring R with R/J(R) Boolean, every endomorphism of M is a sum of an idempotent endomorphism and a locally nilpotent endomorphism. As a consequence, it is proved that, for a vector space V over a division ring D, every linear transformation of V is a sum of an idempotent linear transformation and a locally nilpotent linear transformation if and only if D≅F2. This can be seen as an answer to the “local” version of a question raised by Breaz et al. in [1] on nil-cleanness of the ring of linear transformations of an infinite dimensional vector space.

Self-classification of assembly database using evolutionary method

PurposeThe purpose of the paper is to develop a method of automatic classification of the components of the assembly units. The method is crucial for developing an automatic ship assembly planning tools. The proposed method takes into account the assumptions specific for shipbuilding technology processes: high complexity of structures, difficult expert-based classification of components, fixed priority relations between connections resulting from geometrical constraints and demands of welding processes.Design/methodology/approachThe set of ex post determined liaisons and assembly sequences constitutes the database of structures which have been made-up earlier. The components classification problem is solved using matrix coding of graphs. Information in such form is stored in the database. The minimization of number of cycles in the graph of classes sequence and minimization of diversity of classes within all constructions has been proposed as criteria of optimization. The genetic algorithm has been proposed as a solution method.FindingsThe proposed method solves the problem of components’ classifications. It allows setting the pattern of priorities between classes of various connections. This gives a chance to determine the relationship constraints between the connections of new structures for which assembly sequences are not established.Research limitations/implicationsMathematical formulation of the database is quite laborious. The possibility of partial automation of this process should be considered. Owing to the complexity of the problem, a relatively simple objective function has been proposed. During a ship hull assembly, additional criteria should be taken into account, what will be the direction of further research.Practical implicationsAutomatic classification of components is dedicated for implementation in shipyards and similar assembly systems. Tests performed by the authors confirm efficiency of presented method in supporting management of the database and assembly of new structures planning. Suggested activity-oriented approach allows for easy conversion of any assembly unit structure to the form of a matrix.Originality/valueThe new approach for components classification according to its assembly features distinguishes the proposed method from others. The use of nilpotent matrix theory in an acyclicity of graphs analysis is also a unique achievement. Original crossover and mutation operators for assembly sequence were proposed in the article.

T-branes at the limits of geometry

Singular limits of 6D F-theory compactifications are often captured by T-branes, namely a non-abelian configuration of intersecting 7-branes with a nilpotent matrix of normal deformations. The long distance approximation of such 7-branes is a Hitchin-like system in which simple and irregular poles emerge at marked points of the geometry. When multiple matter fields localize at the same point in the geometry, the associated Higgs field can exhibit irregular behavior, namely poles of order greater than one. This provides a geometric mechanism to engineer wild Higgs bundles. Physical constraints such as anomaly cancellation and consistent coupling to gravity also limit the order of such poles. Using this geometric formulation, we unify seemingly different wild Hitchin systems in a single framework in which orders of poles become adjustable parameters dictated by tuning gauge singlet moduli of the F-theory model.

Matrices over Zhou nil-clean rings

ABSTRACTA ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Let R be a Zhou nil-clean ring. If R is 2-primal (of bounded index), we prove that every square matrix over R is the sum of two tripotents and a nilpotent. These provide a large class of rings over which every square matrix has such decompositions by tripotent and nilpotent matrices.

ВСЕГДА НЕВЫРОЖДЕННЫЕ МНОГОЧЛЕНЫ ОТ ДВУХ ПРОЕКТОРОВ

This paper discusses the polynomials of two projectors that with any selection of these projectors have the value of the nonsingular matrix. Results of work [1] about block-triangular form pair of projectors apply to deduce equations, that the coecients of always nonsingular polynomials satisfy to. From the equations is obtained the main result, namely always nonsigular polynomial can be decomposed into a product of special polynomials. Special polynomial of two projectors P; Q is a linear binomial I + P; I + Q, or a polynomial like this I + x1(PQP PQ) + x2(PQPQP PQPQ) + ::: . It is proved that special polynomials are irreducible. It turns out that linear binomials can be rearranged with some other special polynomials. If in a product of special polynomials the linear binomials are rearranged as much as possible to the left, you will get a product of special polynomials, called standard. It is proved that the standard form of product by special polynomials is unigue. The obtained results have provided a description of the structure of all polynomials of two projectors that with any selection of these projectors are nilpotent matrices (nilpotent polynomials). Similar results were obtained for the involute polynomials and polynomialsprojectors.

Asymptotics of the Jordan Normal Form of a Random Nilpotent Matrix

We study the Jordan normal form of an upper triangular matrix constructed from a random acyclic graph or a random poset. Some limit theorems and concentration results for the number and sizes of Jordan blocks are obtained. In particular, we study a linear algebraic analog of Ulam’s longest increasing subsequence problem.

Linear spaces of symmetric nilpotent matrices

In 1958 Gerstenhaber showed that if L is a subspace of the vector space of the square matrices of order n over some field F, consisting of nilpotent matrices, and the field F is sufficiently large, then the maximal dimension of L is n(n−1)2, and if this dimension is attained, then the space L is triangularizable. Linear spaces of symmetric matrices seem to be first studied by Meshulam in 1989 in view of the bound of their rank. Although it seems unnatural to ask when a linear space of symmetric matrices is made of nilpotents and when it is triangular, we find a way to do so by going to an equivalent notion for symmetric matrices, i.e. persymmetric matrices. We develop a theory that enables us to prove extensions of some beautiful classical triangularizability results to the case of symmetric matrices. Not only the Gerstenhaber's result, but also Engel, Jacobson and Radjavi theorems can be extended. We also study maximal linear spaces of symmetric nilpotents of smaller dimension.

Hydrogen Atom and Equivalent Form of the Lévy-Leblond Equation

We discuss the equivalent form of the Lévy-Leblond equation such that the nilpotent matrices are two-dimensional. We show that this equation can be obtained in the non-relativistic limit of the (2+1)-dimensional Dirac equation. Furthermore, we analyze the case with four-dimensional matrices, propose a Hamiltonian for the equation in (3+1) dimensions, and solve it for a Coulomb potential. The quantized energy levels for the hydrogen atom are obtained, and the result is consistent with the non-relativistic quantum mechanics.

Nilpotent matrices and the minus partial order

In this paper, {1}-inverses of a nilpotent matrix as well as matrices above a given nilpotent matrix under the minus partial order are characterized.

Solving the Yang–Baxter-like matrix equation for nilpotent matrices of index three

ABSTRACTLet A be a nilpotent matrix of index three and consider the Yang–Baxter-like matrix equation AXA=XAX. We first obtain a simplified matrix equation of the same type with A replaced by its Jordan form, whose Jordan blocks are at most . Then we obtain a system of matrix equations of smaller sizes to find all the commuting solutions of the original matrix equation.

Index reduction for rectangular descriptor systems via feedbacks

Index plays a fundamental role in the study of descriptor systems. For regular descriptor systems, calculation of the index can be performed by calculating the index of the nilpotent matrix obtained by means of the Weierstrass canonical form. Notwithstanding, if the system is not regular, there is no algebraic technique to determine the index of the system. A sufficient algebraic criterion is provided to determine the index of a general linear time-invariant descriptor systems. Thereafter, we provide an alternate but lucid proof of the fact that impulse controllability is necessary and sufficient for the existence of a semistate feedback such that the closed loop system is of the index at most one. Finally, a sufficient test for the existence of a semistate feedback such that the closed loop system is of the index at most two is provided. Examples are given to illustrate the presented theory.

On some matrix analogs of the little Fermat theorem

The rings over which every square matrix is representable as a sum of a nilpotent matrix and a q-potent matrix, where q is a positive integer power of a prime, are studied. As consequences, matrix analogs of the little Fermat theorem are obtained.

Nilpotent linear spaces and Albert's Problem

In [8] M.A. Fasoli classified, up to conjugation, all the maximal vector subspaces of M4(C), in which all the elements are nilpotent matrices. This result will allow us to solve Albert's Problem [5] for commutative power-associative C-nilalgebras of dimension n and nilindex n−3 in an affirmative way.

UN-rings

A nonzero ring is called a UN-ring if every nonunit is a product of a unit and a nilpotent element. We show that all simple Artinian rings are UN-rings and that the UN-rings whose identity is a sum of two units (e.g. if 2 is a unit), form a proper class of 2-good rings (in the sense of P. Vámos). Thus, any noninvertible matrix over a division ring is the product of an invertible matrix and a nilpotent matrix.

Explicit generators in rectangular affine $$\mathcal {W}$$ W -algebras of type A

We produce in an explicit form free generators of the affine W-algebra of type A associated with a nilpotent matrix whose Jordan blocks are of the same size. This includes the principal nilpotent case and we thus recover the quantum Miura transformation of Fateev and Lukyanov.

An extension of a theorem of Kaplansky

ABSTRACTA theorem of Kaplansky asserts that a semigroup of matrices with entries from a field whose members all have singleton spectra is triangularizable. Indeed, Kaplansky’s Theorem unifies well-known theorems of Kolchin and Levitzki on simultaneous triangularizability of semigroups of unipotent and nilpotent matrices, respectively. First, for a division ring D of characteristic zero whose center intersects its multiplicative commutator group in a finite group, we prove that the counterpart of Kolchin’s Theorem over D implies that of Kaplansky’s Theorem over D. Next, we note that this proof, when adjusted in the setting of fields, provides a new and simple proof of Kaplansky’s Theorem over fields of characteristic zero. We show that if Kaplansky’s Theorem holds over a division ring D, which is for instance the case over general fields, then a generalization of Kaplansky’s Theorem holds over D, and in particular over general fields.

Nilpotent-invariant modules and rings

ABSTRACTAutomorphism-invariant modules, due to Lee and Zhou, generalize the notion of quasi-injective modules. A module which is invariant under automorphisms of its injective hull is called an automorphism-invariant module. Here we carry out a study of the module which is invariant under nilpotent endomorphisms of its injective envelope, such as modules are called nilpotent-invariant. Many basic properties are obtained. For instance, it is proved that (1) nilpotent-invariant modules have the (C3) property, (2) if is nilpotent-invariant, then M1 and M2 are relative injective. In this paper, we also show that (3) a simple right nilpotent-invariant ring R is either right self-injective or RR is uniform square-free.

Infinite-dimensional features of matrices and pseudospectra

Given a Hilbert space operator T, the level sets of function ΨT(z)=‖(T−z)−1‖−1 determine the so-called pseudospectra of T. We set ΨT to be zero on the spectrum of T. After giving some elementary properties of ΨT (which, as it seems, were not noticed before), we apply them to the study of the approximation. We prove that for any operator T, there is a sequence {Tn} of finite matrices such that ΨTn(z) tends to ΨT(z) uniformly on C. In this proof, quasitriangular operators play a special role. This is merely an existence result, we do not give a concrete construction of this sequence of matrices.One of our main points is to show how to use infinite-dimensional operator models in order to produce examples and counterexamples in the set of finite matrices of large size. In particular, we get a result, which means, in a sense, that the pseudospectrum of a nilpotent matrix can be anything one can imagine. We also study the norms of the multipliers in the context of Cowen–Douglas class operators. We use these results to show that, to the opposite to the function ΨS, the function ‖S−z‖ for certain finite matrices S may oscillate arbitrarily fast even far away from the spectrum.