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Real and complex Schur forms have been receiving increasing attention from the fluid mechanics community recently, especially related to vortices and turbulence. Several decompositions of the velocity gradient tensor, such as the triple decomposition of motion (TDM) and normal-nilpotent decomposition (NND), have been proposed to analyze the local motions of fluid elements. However, due to the existence of different types and non-uniqueness of Schur forms, as well as various possible definitions of NNDs, confusion has spread widely and is harming the research. This work aims to clean up this confusion. To this end, the complex and real Schur forms are derived constructively from the very basics, with special consideration for their non-uniqueness. Conditions of uniqueness are proposed. After a general discussion of normality and nilpotency, a complex NND and several real NNDs as well as normal-nonnormal decompositions are constructed, with a brief comparison of complex and real decompositions. Based on that, several confusing points are clarified, such as the distinction between NND and TDM, and the intrinsic gap between complex and real NNDs. Besides, the author proposes to extend the real block Schur form and its corresponding NNDs for the complex eigenvalue case to the real eigenvalue case. But their justification is left to further investigations. [ABSTRACT FROM AUTHOR]

Let an n × n complex matrix A be such that I + A is invertible. The Cayley transform of A , denoted by C (A) , is defined as C (A) = (I + A) − 1 (I − A). Fallat and Tsatsomeros (2002) [5] and Mondal et al. (2024) [15] studied the Cayley transform of a matrix A in the context of P-matrices, H-matrices, M-matrices, totally positive matrices, positive definite matrices, almost skew-Hermitian matrices, and semipositive matrices. In this paper, the investigation of the Cayley transform is continued for Toeplitz matrices, circulant matrices, unipotent matrices, and dual matrices. An expression of the Cayley transform for dual matrices is established. It is shown that the Cayley transform of a dual symmetric matrix is always a dual symmetric matrix. The Cayley transform of a dual skew-symmetric matrix is discussed. The results are illustrated with examples. [ABSTRACT FROM AUTHOR]

It is shown that the nilpotency of submatrices of a certain class of adjacency matrices is equivalent to the Collatz conjecture. Our main result extends the previous work of Alves et al. and clarifies a conjecture made by Cardon and Tuckfield. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Over any GCD (greatest common divisor) commutative domain we show that the nontrivial 2×22\times 2 idempotent matrices are products of two nilpotent matrices. In order to find explicitly such decompositions, two procedures are described. Assisted by computer, we were able to find an example of idempotent 2×22\times 2 matrix over Z[−5]{\mathbb{Z}}\left[\sqrt{-5}], which shows that the GCD condition is (sufficient but) not necessary. Finally, a generalization is discussed and some open questions stated.

For a given p × q complex matrix C , a necessary and sufficient condition is obtained for the existence of a matrix X satisfying J p X − X J q = C , here J r denotes the r × r Jordan block of 0. An easy construction of the solution X is given if it exists. These results lead to a proof of the fact that a nilpotent matrix is similar to a direct sum of Jordan blocks. [ABSTRACT FROM AUTHOR]

We propose a heuristic method to solve polynomial matrix equations of the type ∑ k = 1 m a k X k = B , where a k 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. [ABSTRACT FROM AUTHOR]

A ring R is feebly nil-clean if for any a ∈ R there exist two orthogonal idem-potents e, f ∈ R and a nilpotent w ∈ R such that a = e − f + w. Let R be a 2-primal feebly nil-clean ring. We prove that every matrix ring over R is feebly nil-clean. The result for rings of bounded index is also obtained. These provide many classes of rings over which every matrix is the sum of orthogonal idempotent and nilpotent matrices. [ABSTRACT FROM AUTHOR]

This work is concerned with determining a general expression for roots of upper and lower triangular matrices with a constant diagonal. Numerical examples of the results are also presented. [ABSTRACT FROM AUTHOR]

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.

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 unitary 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. [ABSTRACT FROM AUTHOR]

An element u of a ring R is called unipotent if u - 1 is nilpotent. Two elements a, b ∈ R are called unipotent equivalent if there exist unipotents p, q ∈ R such that b = q-1ap. Two square matrices A, B are called strongly unipotent equivalent if there are unipotent triangular matrices P, Q with B = Q-1AP. In this paper, over commutative reduced rings, we characterize the matrices which are strongly unipotent equivalent to diagonal matrices. For 2 × 2 matrices over Bézout domains, we characterize the nilpotent matrices unipotent equivalent to some multiples of E12 and the nontrivial idempotents unipotent equivalent to E11. [ABSTRACT FROM AUTHOR]

We will use commutators to provide decompositions of 3 × 3 matrices as sums whose terms satisfy some polynomial identities, and we apply them to bounded linear operators and endomorphisms of free modules of infinite rank. In particular it is proved that every bounded operator of an infinite-dimensional complex Hilbert space is a sum of four automorphisms of order 3 and that every simple ring that is obtained as a quotient of the endomorphism ring of an infinite-dimensional vector space modulo its maximal ideal is a sum of three nilpotent subrings. [ABSTRACT FROM AUTHOR]

For given square matrices A and B we denote by Y = AB−BA and by Z = AY −YA. It is well known that if A and Y commute, i.e., if Z = 0, then Y is a nilpotent matrix. In this note we show that the same is true if Y Z = ZY . We also generalize this result by using commutators of higher order. [ABSTRACT FROM AUTHOR]

In order to find a suitable expression of an arbitrary square matrix over an arbitrary field, we prove that every square matrix over an infinite field is always representable as a sum of a diagonalizable matrix and a nilpotent matrix of order less than or equal to two. In addition, each 2 × 2 matrix over any field admits such a representation. We, moreover, show that, for all natural numbers n ≥ 3, every n × n matrix over a finite field having no less than n + 1 elements also admits such a decomposition. The latter completes a recent example due to Breaz [Matrices over finite fields as sums of periodic and nilpotent elements. Linear Algebra Appl. 2018;555:92–97]. As a consequence of these decompositions, we show that every nilpotent matrix over a field can be expressed as the sum of a potent matrix and a square-zero matrix. This somewhat improves on recent results due to Abyzov et al. [On some matrix analogues of the little Fermat theorem. Mat Zametki. 2017;101(2):163–168] and Shitov [The ring M 8 k + 4 (Z 2) is nil-clean of index four. Indag Math (N.S.). 2019;30:1077–1078]. [ABSTRACT FROM AUTHOR]

We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a 2 × 2 matrix A over a projective-free ring R is strongly J-clean if and only if A ∈ J(M2(R)), or I2 − A ∈ J(M2(R)), or A is similar to ( 0 λ 1 μ ) , where λ ∈ J(R), μ ∈ 1 + J(R), and the equation x2 − xμ − λ = 0 has a root in J(R) and a root in 1 + J(R). We further prove that f(x) ∈ R[[x]] is strongly J-clean if f(0) ∈ R be optimally J-clean. [ABSTRACT FROM AUTHOR]

Over commutative domains we characterize the singular 2 × 2 matrices which are products of two idempotents or products of two nilpotents. The relevant casees are the matrices with zero second row and the singular matrices with only nonzero entries.

It is known that every normal matrix with a simple spectrum can be obtained by a rank-one perturbation of some nilpotent matrix . In this assertion, a normal matrix can actually be replaced by an arbitrary diagonalizable matrix. We determine the possible values of the index of nilpotency of the matrix . [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

In this note, we give a proof that a matrix of determinant 0 on any algebraically closed field is the product of a nilpotent matrix and a unipotent matrix which only uses elementary facts. [ABSTRACT FROM AUTHOR]

Reopening a cold case, Inspector Echelon, high-ranking in the Row Operations Center, is searching for a lost linear map, known to be nilpotent. When a partially decomposed matrix is unearthed, he reconstructs its reduced form, finding it singular. But were its origins nilpotent? [ABSTRACT FROM AUTHOR]

A ring R is nil-clean of index k if there exists a positive integer k such that every element a ∈ R can be expressed as a = e + q where e 2 = e and q k = 0 , and k is the smallest integer with this property. It is known that the matrix ring M n (F 2) is nil-clean of index 3 or 4 for every n ≥ 3 , while the question for which n the nil-clean index is exactly 3 (resp. 4) has not been completely answered yet. In this paper we give a complete answer to this question. It turns out that M n (F 2) is nil-clean of index 3 if and only if n ∈ { 3 , 5 , 6 , 7 , 10 , 11 , 15 }. [ABSTRACT FROM AUTHOR]

Let R be an associative unital ring and not necessarily commutative. We analyze conditions under which every n × n matrix A over R is expressible as a sum A = E 1 + ... + E s + N of (commuting) idempotent matrices E i and a nilpotent matrix N. [ABSTRACT FROM AUTHOR]

In recent years, a lot of research has focused on understanding the behavior of when synchronous and asynchronous phases occur, that is, the existence of chimera states in various networks. Chimera states have wide-range applications in many disciplines including biology, chemistry, physics, or engineering. The object of research in this paper is a coupled map lattice of matrices when each node is described by an iterative map of matrices of order two. A regular topology network of iterative maps of matrices was formed by replacing the scalar iterative map with the iterative map of matrices in each node. The coupled map of matrices is special in a way where we can observe the effect of divergence. This effect can be observed when the matrix of initial conditions is a nilpotent matrix. Also, the evolution of the derived network is investigated. It is found that the network of the supplementary variable $\mu$ can evolve into three different modes: the quiet state, the state of divergence, and the formation of divergence chimeras. The space of parameters of node coupling including coupling strength $\varepsilon$ and coupling range $r$ is also analyzed in this study. Image entropy is applied in order to identify chimera state parameter zones.

Copyright of Lietuvos Matematikos Rinkinys is the property of Vilnius University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We prove that, over a field F of characteristic p > 0 , every nil-clean nonderogatory matrix A with tr (A) ≠ 1 has a decomposition A = E + V such that E 2 = E and V p + 1 = 0. If tr (A) = 1 then we have a similar decomposition with V p + 2 = 0. Consequently, every nilpotent matrix over F is a sum of an idempotent matrix and a nilpotent matrix of nilpotence index at most p + 1. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of the EP-nilpotent decomposition and present some of its applications. By applying the decomposition, we introduce two partial orders (the E-N partial order and the E-S partial order) and characterize their properties. The two partial orders above are non-minus-type partial orders, although the method used to introduce the two partial orders is similar to that of the C-N partial order, a minus-type partial order. [ABSTRACT FROM AUTHOR]

A ring R is Zhou nil-clean if every element in R is the sum of a nilpotent and two tripotents. Let R be a Zhou nil-clean ring. If R is of bounded index or 2-primal, we prove that every square matrix over R is the sum of a nilpotent and a linear combination of two idempotents. This provides a large class of rings over which every square matrix has such decompositions by nilpotent and linear combination of idempotent matrices. [ABSTRACT FROM AUTHOR]

Let k be an integer such that k ≥ 2. An n-by-n matrix A is said to be strictly k-zero if Ak = 0 and Am ≠ 0 for all positive integers m with m < k. Suppose A is an n-by-n matrix over a field with at least three elements. We show that, if A is a nonscalar matrix with zero trace, then (i) A is a sum of four strictly k-zero matrices for all k ∈{2,..., n}; and (ii) A is a sum of three strictly k-zero matrices for some k ∈{2,..., n}. We prove that, if A is a scalar matrix with zero trace, then A is a sum of five strictly k-zero matrices for all k ∈{2,..., n}. We also determine the least positive integer m, such that every square complex matrix A with zero trace is a sum of m strictly k-zero matrices for all k ∈{2,..., n}.

An approximated empirical mode decomposition generates a set of approximated intrinsic mode functions via a linear, nonadaptive but iterative approach. The decomposition was found to be very useful for a content-independent pattern recognition application. As the process is characterized by a system kernel matrix and performed iteratively, the approximated intrinsic mode functions can be understood as the original signals processed by a set of mask operations. Here, some properties of the decomposition are studied and an optimal design of the system kernel matrix is proposed. It is found that there is only one approximated intrinsic mode function if the exact perfect reconstruction condition is satisfied. Obviously, the approximated intrinsic mode function is the original signal. Therefore, the decomposition is practically not meaningful. To address this issue, the infinite number of iterations in the algorithm is truncated to a finite number of iterations. Also, the design of the system kernel matrix is formulated as an optimization problem. In particular, the exact perfect reconstruction error between the sum of the approximated intrinsic mode functions and the original signal is minimized and the total absolute sum of the difference between any two different eigenvalues of the iterative matrix is maximized subject to a stability condition. Here, the stability condition refers to the eigenvalues of the iterative matrix being between zero and one. Since the optimization problem is nonsmooth and nonconvex, a genetic algorithm is employed for finding its near global optimal solution. Compared to the conventional approximated empirical mode decomposition, computer numerical simulations show that our proposed approach can achieve more than one approximated intrinsic mode function with each approximated intrinsic mode function corresponding to an output of a more meaningful mask operation. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Harbin University of Science & Technology is the property of Journal of Harbin University of Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we give a necessary and sufficient condition for the powers of an interval matrix to be nilpotent. We show an interval matrix A is nilpotent if and only if p(B) = 0, where B is a point matrix, introduced by Mayer (Linear Algebra Appl. 58 (1984) 201-216), constructed by the (*) property. We observed that the spectral radius, determinant, and trace of a nilpotent interval matrix equal zero but in general its converse is not true. Some properties of nonnegative nilpotent interval matrices are derived. We also show that an irreducible interval matrix A is nilpotent if and only if |A| is nilpotent. [ABSTRACT FROM AUTHOR]

Abstract Consider the special linear Lie algebra sl n (K) over an infinite field of characteristic different from 2. We prove that for any nonzero nilpotent X there exists a nilpotent Y such that the matrices X and Y generate the Lie algebra sl n (K). [ABSTRACT FROM AUTHOR]

We prove that every n × n matrix M over a field of odd cardinality q has a decomposition of the form M = E + N such that E q = E and N 3 = 0 . [ABSTRACT FROM AUTHOR]

In this paper, we give several characterizations of complex nilpotent matrices of order 2, which are based on the Cayley-Hamilton theorem. In terms of traces and determinants of matrices, we describe the conditions on A, B 2 M2(C) under which the commutator AB - BA is a nilpotent matrix. [ABSTRACT FROM AUTHOR]

A 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]