Nilpotent Matrices Research Articles

A new extension of the core inverse

There are a number of extensions of the core inverse represented by the products of known generalized inverses, like the core-EP inverse, DMP inverse, GC inverse and so on. However, none of them is an inner inverse of the matrix, and especially for an arbitrary nilpotent matrix, such inverses are always null. Our goal is to use a new technic to introduce an extension of the core inverse that preserves the interesting property of being an inner inverse of the matrix which need not necessarily be the null matrix of a nilpotent matrix. We define the extended core inverse for square complex matrices combining the sum and the difference of three known generalized inverses. Various properties and representations of the extended core inverse are developed. The extended dual core inverse is investigated too. We apply the extended core inverse to solve some systems of linear equations and one minimization problem. A significant normal equation related to least-squares solutions can be solved using the extended core inverse.

On separability of semigroups: application to models of physics and discrete optimization

Nilpotent semigroups are used in various fields of physics to model processes like system decay in quantum mechanics, state transitions in statistical mechanics, and simplifying dynamical systems. They also assist in optimizing control processes. The set of nilpotent matrices forms a semigroup and plays a key role in Jordan decomposition, which has many physical applications. In quantum mechanics, the Jordan form helps analyze linear operators on Hilbert spaces, especially with systems involving eigenvalue multiplicities. In vibration analysis, it aids in studying normal modes of mechanical systems with degenerate frequencies. Additionally, Jordan decomposition simplifies control theory for linear time-invariant systems, stability analysis in dynamical systems, and the behavior of coupled oscillators, as well as solving systems of linear differential equations. Certainly, the semigroup theory has even more applications in theoretical computer science: suffice it to say, for example, that some authors identify semigroups and finite automata. In this paper, we address the problem of P-separability of semigroups with respect to three key predicates: equality-separability, divisibility-separability, and subsemigroup-separability, achieved through homomorphisms into a nilpotent semigroup. We present some significant results that advance the understanding of these concepts.

Expressing Partial Order-Preserving Transformations as Products of Nilpotents

Let S be a semigroup with zero, then an element a∈S is called nilpotent, if there exists a positive integer n such that . In the partial transformation semigroup on , where is a non-empty set, denoted by , the empty map is the zero of . Further, nilpotency is a structural property of fundamental importance which arises in many situations, for instance in linear algebra, we have nilpotent matrices and much more in other branches of mathematics. In this study, we will investigate, some elements of semigroups of partial one-to-one order-preserving transformations (denoted by ) and partial order-preserving transformations (denoted by ), and some basic properties of nilpotent transformations. We have studied certain properties of nilpotent elements in , symmetric inverse semigroup, where is finite and infinite, certain properties of nilpotent elements in , descriptions of the elements of 〈N〉, where N is the set of nilpotents of and we used the same letter N to denote that of , and used them to establish the following results, in this paper (that is, we have used the knowledge that we have acquired to establish the following results): - a partial ono-to-one order-preserving transformation , has from 1 to fixed, and there is an upper jump of length two(2) between and , and lower jumps of length one(1) and one(1), immediately after , can be expressed as a product of three nilpotents. – a partial one-to-one order-preserving transformation , has from 1 to fixed, and there is an upper jump of length three (3) between and and lower jumps of length two (2) and one (1), immediately after , can be expressed as a product of fewer than three nilpotents. – a partial order-preserving transformation , has from 1 to fixed, but belongs to , and there is an upper jump of length two(2) between and , and lower jumps of length two(2) and one(1), immediately after , cannot be expressed as a product of fewer than three nilpotents. – a partial order-preserving transformation , has from 1 to fixed, but belongs to , and there is an upper jump of length three (3) between and , and lower jumps of length three (3) and one (1), immediately after n-5, can be expressed as a product of fewer than three nilpotents. – a transformation , partial transformation semigroup, is not nilpotent if and only if the graph of the transformation has a loop, and - a transformation , partial one-to-one transformation semigroup, which has a nilpotent path and a permutation part is not nilpotent

Isomorphisms between lattices of hyperinvariant subspaces

Given two nilpotent endomorphisms, we determine when their lattices of hyperinvariant subspaces are isomorphic. The study of the lattice of hyperinvariant subspaces can be reduced to the nilpotent case when the endomorphism has a Jordan-Chevalley decomposition; for example, it occurs if the underlying field is the field of complex numbers.

2-Products of Idempotent by Nilpotent Matrices

Over Prüfer domains, we characterize idempotent by nilpotent 2-products of 2×2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\ imes 2$$\\end{document} matrices. Nilpotents are always such products. We also provide large classes of rings over which every 2×2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\ imes 2$$\\end{document} idempotent matrix is such a product. Finally, for 2×2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\ imes 2$$\\end{document} matrices over GCD domains, idempotent–nilpotent products which are also nilpotent–idempotent products are characterized.

A note on splittable linear Lie algebras

A linear Lie algebra is splittable if it contains the semisimple and nilpotent parts of each element. It is early known that a solvable linear Lie algebra g is splittable if and only if g=a+n, where a is an abelian subalgebra of g composed of semisimple elements and n is the ideal of all nilpotent matrices of g. In this paper, using elementary linear algebra we give a direct proof of the theorem and related results. Besides, we determine the structure of linear Lie algebras composed of semisimple or nilpotent elements.

A Heuristic Method for Solving Polynomial Matrix Equations

We propose a heuristic method to solve polynomial matrix equations of the type ∑k=1makXk=B, where ak are scalar coefficients and X and B are square matrices of order n. The method is based on the decomposition of the B matrix as a linear combination of the identity matrix and an idempotent, involutive, or nilpotent matrix. We prove that this decomposition is always possible when n=2. Moreover, in some cases we can compute solutions when we have an infinite number of them (singular solutions). This method has been coded in MATLAB and has been compared to other methods found in the existing literature, such as the diagonalization and the interpolation methods. It turns out that the proposed method is considerably faster than the latter methods. Furthermore, the proposed method can calculate solutions when diagonalization and interpolation methods fail or calculate singular solutions when these methods are not capable of doing so.

Jordan structures of nilpotent matrices in the centralizer of a nilpotent matrix with two Jordan blocks of the same size

In this paper we characterize all nilpotent orbits under the action by conjugation that intersect the nilpotent centralizer of a nilpotent matrix B consisting of two Jordan blocks of the same size. We list all the possible Jordan canonical forms of the nilpotent matrices that commute with B by characterizing the corresponding partitions.

Subspaces fixed by a nilpotent matrix

Subspaces fixed by a nilpotent matrix

On the Similarity over the Ring of Integers of Certain Nilpotent Matrices of Maximal Rank

This paper is devoted to the problem of matrix similarity recognition over the ring of integers for some families of matrices. Namely, nilpotent upper triangular matrices of maximal rank are considered such that only first and second superdiagonals of these matrices are non-zero. Several necessary conditions are obtained for similarity of such matrices to matrices of the form superdiag(a1,a2,…,an−1) with a single non-zero superdiagonal, that is a generalization of the Jordan cell Jn(0)=superdiag(1,1,…,1) . These conditions are formulated in simple terms of divisibility and greatest common divisors of matrix elements. The result is obtained by reducing the problem of similarity recognition to the problem of solving in integers a system of linear equations and applying the known necessary similarity conditions for arbitrary matrices. Under some additional conditions on the elements a1,a2,…,an−1 of the first superdiagonal of matrix A , it is proven that the matrix A is similar to matrix superdiag(a1,a2,…,an−1) regardless of the values of the elements of the second superdiagonal. Moreover, for the considered matrices of the third and the fourth orders, easily verifiable necessary and sufficient similarity conditions are obtained describing their similarity to a matrix of the form superdiag(a1,a2,…,an−1) .

Real supersymmetric solutions of (3,2) signature five-dimensional supergravity

We classify supersymmetric solutions of D = 5 (3,2) signature supergravity with either vanishing or imaginary gauge coupling constant preserving the minimal N = 2 supersymmetry. We prove that the geometry of such solutions is characterized by a nilpotent integrable endomorphism, and obtain the necessary and sufficient conditions on the fluxes imposed by supersymmetry. We also construct examples of supersymmetric domain wall solutions for which the nilpotent integrable endomorphism is associated with a Killing–Yano two-form, as well as a new descendant ‘preon’ solution which preserves N = 6 supersymmetry. This is notable as such N = 6 descendant solutions do not exist in the standard signature D = 5 supergravity.

The solution of the Loewy–Radwan conjecture

A seminal result of Gerstenhaber gives the maximal dimension of a linear space of nilpotent matrices. It also exhibits the structure of such a space when the maximal dimension is attained. Extensions of this result in the direction of linear spaces of matrices with a bounded number of eigenvalues have been studied. In this paper, we answer what is perhaps the most general problem of the kind as proposed by Loewy and Radwan, by solving their conjecture in the positive. We give the maximal dimension of a vector space of n × n matrices with no more than k<n eigenvalues. We also exhibit the structure of the spaces for which this dimension is attained.

The Wasserstein metric matrix and its computational property

By further exploring and deeply analyzing the concrete algebraic structures and essential computational properties about the Wasserstein-1 metric matrices of one- and two-dimensions, we show that they can be essentially expressed by the Neumann series of nilpotent matrices and, therefore, the products of these matrices with a prescribed vector can be accomplished accurately and stably in the optimal computational complexities through solving unit bidiagonal triangular systems of linear equations. We also give appropriate generalizations of these one- and two-dimensional Wasserstein-1 metric matrices, as well as their corresponding extensions to higher dimensions, and demonstrate the algebraic structures and computational properties of these generalized and extended Wasserstein-1 metric matrices.

Higher order derivatives of the adjugate matrix and the Jordan form

In this short note, we show that the higher-order derivatives of the adjugate matrix Adj(z−A), are related to the nilpotent matrices and projections in the Jordan decomposition of the matrix A. These relations appear as a factorization of the derivative of the adjugate matrix as a product of factors related to the eigenvalues, nilpotent matrices and projectors. The novel relations are obtained using the Riesz projector and functional calculus. The results presented here can be considered to be a generalization of Thompson and McEnteggert's theorem relating the adjugate matrix to the orthogonal projection on the eigenspace of simple eigenvalues for symmetric matrices. They can also be seen as a complement to some earlier results by B. Parisse, M. Vaughan that relate derivatives of the adjugate matrix to the invariant subspaces associated with an eigenvalue. Our results can also be interpreted as a general eigenvector-eigenvalue identity. Many previous works have dealt with relations between the projectors on the eigenspaces and the derivatives of the adjugate matrix with the characteristic spaces but it seems that there is no explicit mention in the literature of the factorization of the higher-order derivatives of the adjugate matrix as a matrix multiplication involving nilpotent and projector matrices, which appear in the Jordan decomposition theorem.

Nonsurjective zero product preservers between matrix spaces over an arbitrary field

A map Φ between matrices is said to be zero product preserving if Φ ( A ) Φ ( B ) = 0 whenever AB = 0. In this paper, we give concrete descriptions of an additive/linear zero product preserver Φ : M n ( F ) → M r ( F ) between matrix algebras of different dimensions over an arbitrary field F , and n ≥ 2 . In particular, we show that if Φ is linear and preserves zero products then Φ ( A ) = S ( R 1 ⊗ A 0 0 Φ 0 ( A ) ) S − 1 , for some invertible matrices R 1 in M k ( F ) , S in M r ( F ) and a zero product preserving linear map Φ 0 : M n ( F ) → M r − nk ( F ) into nilpotent matrices. If Φ ( I n ) is invertible, then Φ 0 is vacuous. In general, the structure of Φ 0 could be quite arbitrary, especially when Φ 0 ( M n ( F ) ) has trivial multiplication, i.e. Φ 0 ( X ) Φ 0 ( Y ) = 0 for all X, Y in M n ( F ) . We show that if Φ 0 ( I n ) = 0 or r − nk ≤ n + 1 , then Φ 0 ( M n ( F ) ) indeed has trivial multiplication. More generally, we characterize subspaces V of square matrices satisfying XY = 0 for any X , Y ∈ V . Similar results for double zero product preserving maps are obtained.

Decompositions of matrices into a sum of invertible matrices and matrices of fixed nilpotence

For any $n\ge 2$ and fixed $k\ge 1$, we give necessary and sufficient conditions for an arbitrary nonzero square matrix in the matrix ring $\mathbb{M}_n(\mathbb{F})$ to be written as a sum of an invertible matrix $U$ and a nilpotent matrix $N$ with $N^k=0$ over an arbitrary field $\mathbb{F}$.

Discrete fixed-time observers over sensor networks with unknown noise

This paper investigates fixed-time observers for a class of sensor networks with unknown noise. Specifically, the strongly observable condition and observer matching condition are provided for the convenience of analysis. Two kinds of fixed-time observers for sensor networks are developed based on the system’s observability and the properties of nilpotent matrices. The convergence time of the first kind of observer depends on the target state’s dimension and the rank of the noise input matrix. Besides, to further reduce the convergence time, another kind of observer is designed with the thought of the reduced-order observer. Its convergence time is related to the dimension of the target state and the rank of the augmented measurement matrix. Finally, the effectiveness of the proposed fixed-time observers is demonstrated by simulation examples.

Limiting mixed Hodge structures on the relative log de Rham cohomology groups of a projective semistable log smooth degeneration

We prove that the relative log de Rham cohomology groups of a projective semistable log smooth degeneration admit a natural limiting mixed Hodge structure. More precisely, we construct a family of increasing filtrations and a family of nilpotent endomorphisms on the relative log de Rham cohomology groups and show that they satisfy a part of good properties of a nilpotent orbit in several variables.

Hyper Coupled Map Lattices for Hiding Multiple Images

The concept of a hyper coupled map lattice (CML) is presented in this paper. The complexity of the lattice is increased not by adding another spatial dimension of the lattice but by replacing scalar nodal variables by multidimensional square matrices of iterative variables. The proposed scheme exploits the nonlinear effects of the spatiotemporal divergence induced by nilpotent nodal matrices to generate separate secret images at different discrete moments of time during the evolution of the CML. The time variable plays the primary role in the decoding stage of the scheme. The carrying capacity of the proposed scheme is n − 1 different dichotomous digital images, where n is the dimension of the nilpotent nodal matrices. Computational experiments are used to demonstrate the efficacy of the proposed scheme.

Theta divisor and Abel map for 4-by-4 matrices

We consider a hyperbolic quartic curve F(x,y,z)=0 associated with a 4-by-4 nilpotent matrix which has one ordinary double point satisfying F(x,y,z)=F(x,−y,z). Let R be the Riemann surface determined as a non-singular model of the curve F(x,y,z)=0. In this situation, the genus of R is 2. We express the period matrix of R using explicit line integrals depending on a simple symplectic transformation of the canonical cycles which Maple algcurves package provided, and determine the Riemann constant of the Riemann vanishing theorem. We also provide a pictorial visualization of the real parts of the image of an Abel map and theta divisor.