Clément de Seguins Pazzis
We study which matrices are sums of idempotents over a field of non-zero characteristic; in particular, we prove that any such matrix, provided it is large enough, is actually a sum of five idempotents, and even of four when the field is a prime one.
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Motivated by Hirano-Tominaga's work \cite{HT} on rings for which every element is a sum of two idempotents and by de Seguins Pazzis's results \cite{de} on decomposing every matrix over a field of positive characteristic as a sum of idempotent matrices, we address decomposing every matrix over a commutative ring as a sum of three idempotent matrices and, respectively, as a sum of three involutive matrices.
In this paper we provide concrete constructions of idempotents to represent typical singular matrices over a given ring as a product of idempotents and apply these factorizations for proving our main results. We generalize works due to Laffey (Products of idempotent matrices. Linear Multilinear A. 1983) and Rao (Products of idempotent matrices. Linear Algebra Appl. 2009) to noncommutative setting and fill in the gaps in the original proof of Rao's main theorems. We also consider singular matrices over Bézout domains as to when such a matrix is a product of idempotent matrices.
We characterize the idempotent stable range one $2\times 2$ matrices over commutative rings and in particular, the integral matrices with this property. Several special cases and examples complete the subject.
We investigate invertible matrices over finite additively idempotent semirings. The main result provides a criterion for the invertibility of such matrices. We also give a construction of the inverse matrix and a formula for the number of invertible matrices.
Let $R$ be a commutative additively idempotent semiring. In this paper, some properties and characterizations for permanents of matrices over $R$ are established, and several inequalities for permanents are given. Also, the adjiont matrices of matriecs over $R$ are considered. Partial results obtained in this paper generalize the corresponding ones on fuzzy matrices, on lattice matrices and on incline matrices.
A Note on Idempotent Matrices: The Poset Structure and The Construction
本文从偏序结构与构造两方面研究除环及主理想整环上的幂等矩阵，利用块分解刻画偏序，借助Smith标准形给出等价条件，并推广至唯一分解整环及Kronecker积等构造。
Here we characterize the linear operators that preserve rank of matrices over additively idempotent and multiplicatively cancellative semirings. The main results in this article generalize the corresponding results on the two element Boolean algebra and on the max algebra; and holds on max-plus algebra and some other tropical semirings.
Products of Nilpotent and Idempotent Matrices over Finite Local Rings
本文研究有限交换局部主环R上的2阶矩阵，证明幂零与幂等矩阵的乘积必为幂等或幂零矩阵的乘积，IN-与NI-矩阵类重合，还给出其元素数量的显式公式。
Idempotent factorization on some matrices over quadratic integer rings
本文针对Cossu和Zanardo关于二次整数环上特定奇异矩阵幂等分解的猜想，构造了支持与反对该猜想的矩阵类，并证明存在不能分解为两个幂等矩阵乘积的此类矩阵。
Given an arbitrary field K and non-zero scalars a and b, we give necessary and sufficient conditions for a matrix A in M_n(K) to be a linear combination of two idempotents with coefficients a and b. This extends results previously obtained by Hartwig and Putcha in two ways: the field K considered here is arbitrary (possibly of characteristic 2), and the case a is different from b and -b is taken into account.
Let $k = \mathbb{Q}(\sqrtα)$ be a real quadratic number field, where $α$ is a positive square-free integer. Let $\mathcal{O}_k$ be the ring of integers of $k$. In this paper, we prove that a certain set of $2 \times 2$ singular matrices with entries in $\mathcal{O}_k$ can be written as a product of a bounded number of idempotent matrices. Our main theorem can be viewed as a generalization of the recent result by Cossu and Zanardo, which studies finite generation of certain singular matrices by idempotent matrices over $\mathcal{O}_k$ instead of bounded generation of certain singular matrices b…
Decomposition of a square matrix into a product of a nilpotent matrix and an involutive or idempotent matrix
本研究以不变因子为依据，完整刻画了交换域上n阶方阵可分解为幂零矩阵与对合矩阵或幂等矩阵乘积的条件。
Decomposition of matrices into product of idempotents and separativity of regular rings
本文研究正则环上分块矩阵的幂等分解，证明分离正则环和GE环上矩阵可分解为幂等矩阵乘积，并探讨非分离正则环的存在性及全非负矩阵的幂等分解问题。
We explore Jordan derivations of triangular matrices with entries from an additively idempotent semiring. The main result states that for any matrix A over additively idempotent semiring, if we put all the elements of the family of dense submatrices of A to be zeroes, we find a derivative of A. The set of derivations of this type is established.
Idempotent & Nilpotent Operators and Matrices in Bicomplex Space
本文研究双复空间中幂等和幂零算子的性质，定义相关矩阵并推导结果，提出若干定理以确定其存在性和行为条件。
Idempotent factorizations of singular $2\times 2$ matrices over quadratic integer rings
该研究聚焦二次整数环上奇异2×2矩阵的幂等因子分解问题，利用$SL_2(D)$由初等矩阵生成的结论，证明d>0时含零行/零列的矩阵及列-行矩阵均存在幂等分解，补充了d<0时存在不可分解矩阵的相关结论。
Idempotent factorization of matrices over a Prüfer domain of rational functions
本文针对Prüfer非Bézout域的公开问题，研究了有理函数极小Dress环（一种Prüfer域上2×2奇异矩阵的幂等分解问题，给出了保证其可分解为幂等矩阵乘积的若干条件。
We study analogues of classical inequalities for the eigenvalues of sums of pseudo-Hermitian matrices.
Rank Of bicomplex matrices and system of algebraic equations
该研究聚焦双复数矩阵的秩与方程组问题，分析了秩与幂等行/列秩的关系，提出新的求解技术，建立了解存在的充要条件并推导相关结果，为双复数空间的代数问题研究提供了理论支撑。
In a recent article, we gave a full characterization of matrices that can be decomposed as a linear combination of two idempotents with prescribed coefficients. In this one, we use those results to improve on a recent theorem of V. Rabanovich: we establish that every square matrix is a linear combination of three idempotents (for an arbitrary coefficient field rather than just a field of characteristic 0).
Which matrices can be written as sums or products of square-zero matrices? This question is the central premise of this dissertation. Over the past 25 years a significant body of research on products and linear combinations of square-zero matrices has developed, and it is the aim of this study to present this body of research in a consolidated, holistic format, that could serve as a theoretical introduction to the subject. The content of the research is presented in three parts: first results within the broader context of sums and products of nilpotent matrices are discussed, then products of …
We study sums of $k$-potent matrices. We show the conditions by which a complex matrix $A$ can be expressed as a sums of $k$-potent matrices. Also we obtain conditions by which a complex matrix $A$ can be expressed as a sum of finite order elements. This generalize some results obtain by Wu. Also we study the sum of $k$-potent matrices in $\mathcal{M}_{\mathcal{C}f}(F)$ with $F$ be a field and proof that any matrix in this space can be expressed as a sum of $14$ $(k+1)$-potent matrices preserving the result obtain by Slowik.
Factorizations into idempotent factors of matrices over Prüfer domains
针对刻画满足幂等矩阵分解性质的整环这一经典问题，研究证明满足(ID2)的整环必为满足特定可逆矩阵分解条件的Prüfer域，并验证了一大类平面曲线坐标环与Int(Z)符合相关猜想的等价形式。
There is a well known correspondence between the triangle inequality for a distance function on a finite set, and idempotency of an associated matrix over the tropical semiring. Recent research has shed new light on the structure (algebraic, combinatorial and geometric) of tropical idempotents, and in this paper we explore the consequences of this for the metric geometry of tropical polytopes. We prove, for example, that every n-point metric space is realised by the Hilbert projective metric on the vertices of a pure n-dimensional tropical polytope in tropical n-space. More generally, every n-…
Idempotent-free non-solvable evolution algebras over $\mathbb{C}$
本文构造了最小维数（三维）复数演化代数反例，反驳了“可解等价于无非零幂等元”的猜想，并证明了稳定项为演化代数时该等价性成立。
It is known that every complex trace-zero matrix is the sum of four square-zero matrices, but not necessarily of three such matrices. In this note, we prove that for every trace-zero matrix $A$ over an arbitrary field, there is a non-negative integer $p$ such that the extended matrix $A \oplus 0_p$ is the sum of three square-zero matrices (more precisely, one can simply take $p$ as the number of rows of $A$). Moreover, we demonstrate that if the underlying field has characteristic $2$ then every trace-zero matrix is the sum of three square-zero matrices. We also discuss a counterpart of the la…
I provide an alternate and self-contained proof of Botha's theorem on products of idempotent and square-zero matrices where the product contains two square-zero factors, and provide a conclusive characterisation of products of singular quadratic matrices based on previous results.
Nonderogatory matrices as sums of $p$-potent and nilpotent matrices
本文证明在正奇数特征域上，非退化矩阵若可分解为$p$-幂等与幂零矩阵之和，则存在满足$E^p=E$和$V^3=0$的分解。
Minimal Prüfer-Dress rings and products of idempotent matrices
该研究探讨极小Dress环的结构性质，证明部分域上的极小Dress环为赋值或Bézout整环，刻画了$\mathbb{R}(X)$上极小Dress环的元素与理想，并研究其上2×2幂等矩阵乘积问题。
Companion matrices as sums of $p$-potent and nilpotent matrices
本文证明了在特征为奇数 $p$ 的域上，友矩阵能分解为 $p$-幂等矩阵与幂零矩阵之和，当且仅当其迹为单位元的整数倍。
Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived since a quantum logic gates is represented by, or is said to be, a unitary matrix. An efficient algorithm for expressing an idempotent as a unique sum of rank $1$ idempotents with increasing initial zeros is derived. This is used to derive a unique form for mixed matrices. A number of (further) applications are given: for example (i) $U$ is a symmetric unitary matrix if and only if…
Let $A_1,\ldots,A_s$ be unitary commutative rings which do not have non-trivial idempotents and let $A=A_1\oplus\cdots\oplus A_s$ be their direct sum. We describe all idempotents in the $2\times 2$ matrix ring $M_2(A[[X]])$ over the ring $A[[X]]$ of formal power series with coefficients in $A$ and in arbitrary set of variables $X$. We apply this result to the matrix ring $M_2({\mathbb Z}_n[[X]])$ over the ring ${\mathbb Z}_n[[X]]$ for an arbitrary positive integer $n$ greater than 1. Our proof is elementary and uses only the Cayley-Hamilton theorem (for $2\times 2$ matrices only) and, in the s…
An element $a$ in a ring $R$ is strongly J-clean if it is the sum of an idempotent and an element in the Jacobson radical that commutes. We characterize the strongly J-clean $2\times 2$ matrices over 2-projective-free non-commutative rings.
Motivated by a work of Boros, Brualdi, Crama and Hoffman, we consider the sets of (i) possible Perron roots of nonnegative matrices with prescribed row sums and associated graph, and (ii) possible eigenvalues of complex matrices with prescribed associated graph and row sums of the moduli of their entries. To characterize the set of Perron roots or possible eigenvalues of matrices in these classes we introduce, following an idea of Al'pin, Elsner and van den Driessche, the concept of row uniform matrix, which is a nonnegative matrix where all nonzero entries in every row are equal. Furthermore,…
The purpose of this paper is to study the structure of Lotka-Volterra algebras, the set of their idempotent elements and their group of automorphisms. These algebras are defined through antisymmetric matrices and they emerge in connection with biological problems and Lotka-Volterra systems for the interactions of neighboring individuals.
An element of a ring $R$ is called strongly $J^{\#}$-clean provided that it can be written as the sum of an idempotent and an element in $J^{\#}(R)$ that commute. We characterize, in this article, the strongly $J^{\#}$-cleanness of matrices over projective-free rings. These extend many known results on strongly clean matrices over commutative local rings.
We study the multiplication operation of square matrices over lattices. If the underlying lattice is distributive, then matrices form a semigroup; we investigate idempotent and nilpotent elements and the maximal subgroups of this matrix semigroup. We prove that matrix multiplication over nondistributive lattices is antiassociative, and we determine the invertible matrices in the case when the least or the greatest element of the lattice is irreducible.
Additively idempotent matrix semirings
本研究针对加法幂等半环上的矩阵半环，分别刻画交换/有限半环下矩阵半环为同余单半环、几乎整半环及伪MV-代数导出半环下矩阵半环为次直不可约半环的条件。
An element $a\in R$ is provided that there exists an idempotent $e\in R$ such that $a-e\in U(R), ae=ea$ and $eae\in J(eRe)$. In this article, we investigate strongly rad-clean matrices over a commutative local ring. We completely determine when a $2\times 2$ matrix over a commutative local ring is strongly rad-clean.
On sums and products of diagonalizable matrices over division rings
本文将 Botha 关于域上矩阵的和与积分解推广到非交换除环，证明矩阵可表为两个可对角化矩阵之和或积，并据此得到矩阵的 Waring 型结果。
Sums of two nilpotent quaternionic matrices
本文刻画了四元数除环上可表示为两个幂零矩阵之和的方阵，给出了简洁的充要条件，并讨论了n≥3时的特殊情况。
Matrices over finite fields of odd characteristic as sums of diagonalizable and square-zero matrices
本文证明在有限域奇特征下，当域大小≥5时，所有矩阵可分解为可对角化矩阵与平方零矩阵之和，而对于F_3，非退化矩阵及特定不变因子矩阵有不同分解条件。
A bound for the nilpotence index associated to $m$-nil-clean nonderogatory matrices
研究在特定条件下，$n\times n$非奇异矩阵$A$的分解问题，通过证明得出存在$m$个幂等矩阵和一个幂零矩阵使$A$可分解，且给出幂零矩阵幂零指数及$n > p$时$A$按$n$奇偶性的不同分解形式。
Gerstenhaber's theorem states that the dimension of the unital algebra generated by two commuting $n\times n$ matrices is at most $n$. We study the analog of this question for positive matrices with a positive commutator. We show that the dimension of the unital algebra generated by the matrices is at most $\frac{n(n+1)}{2}$ and that this bound can be attained. We also consider the corresponding question if one of the matrices is a permutation or a companion matrix or both of them are idempotents. In these cases, the upper bound for the dimension can be reduced significantly. In particular, th…
Let K be an arbitrary field, and a,b,c,d be elements of K such that the polynomials t^2-at-b and t^2-ct-d are split in K[t]. Given a square matrix M with entries in K, we give necessary and sufficient conditions for the existence of two matrices A and B such that M=A+B, A^2=a A+bI_n and B^2=c B+dI_n. Prior to this paper, such conditions were known in the case b=d=0, a<>0 and c<>0, and also in the case a=b=c=d=0. Here, we complete the study, which essentially amounts to determining when a matrix is the sum of an idempotent and a square-zero matrix. This generalizes results of Wang to an arbitra…
An $R$-module $V$ over a semiring $R$ lacks zero sums (LZS) if $ x +y = 0 \; \Rightarrow \; x = y = 0$. More generally, asubmodule $W$ of $V$ is "summand absorbing", if $ \forall \, x, y \in V: \ x + y \in W \; \Rightarrow \; x \in W, \; y \in W. $ These relate to tropical algebra and modules over idempotent semirings, as well as modules over semirings of sums of squares. In previous work, we have explored the lattice of summand absorbing submodules of a given LZS module, especially those that are finitely generated, in terms of the lattice-theoretic Krull dimension. In this note we describe t…
An element $a\in R$ is very clean provided that there exists an idempotent $e\in R$ such that $ae=ea$ and either $a-e$ or $a+e$ is invertible. A ring $R$ is very clean in case every element in $R$ is very clean. We explore the necessary and sufficient conditions under which a triangular $2\times 2$ matrix ring over local rings is very clean. The very clean $2\times 2$ matrices over commutative local rings are completely determined. Applications to matrices over power series are also obtained.
Matrices over Finite Fields of Characteristic 2 as Sums of Diagonalizable and Square-Zero Matrices
研究特征为2的有限域上矩阵是否可分解为可对角化矩阵与指数不超过2的nilpotent矩阵之和，并在域元素多于三个时完全解决此问题，同时证明F_2上矩阵可表示为指数不超过4的potent矩阵与指数不超过2的nilpotent矩阵之和。
An incline is an additively idempotent semiring in which the product of two elements is always less than or equal to either factor. By making use of prime numbers, this paper proves that A^{11} is less than or equal to A^5 for all 3x3 matrices A over an arbitrary commutative incline, thus giving an answer to an open problem "For 3x3 matrices over any incline (even noncommutative) is X^5 greater than or equal to X^{11}?", proposed by Cao, Kim and Roush in a monograph Incline Algebra and Applications, 1984.
In this paper we deal with the problem of computing the sum of the $k$-th powers of all the elements of the matrix ring $\mathbb{M}_d(R)$ with $d>1$ and $R$ a finite commutative ring. We completely solve the problem in the case $R=\mathbb{Z}/n\mathbb{Z}$ and give some results that compute the value of this sum if $R$ is an arbitrary finite commutative ring $R$ for many values of $k$ and $d$. Finally, based on computational evidence and using some technical results proved in the paper we conjecture that the sum of the $k$-th powers of all the elements of the matrix ring $\mathbb{M}_d(R)$ is alw…
We give an elementary proof of a Caratheodory-type result on the invertibility of a sum of matrices, due first to Facchini and Barioli. The proof yields a polynomial identity, expressing the determinant of a large sum of matrices in terms of determinants of smaller sums. Interpreting these results over an arbitrary commutative ring gives a stabilization result for a filtered family of ideals of determinants. Generalizing in another direction gives a characterization of local rings. An analogous result for semilocal rings is also given -- interestingly, the semilocal case reduces to the case of…
We investigate the possibility of constructing Kochen-Specker uncolorable sets of idempotent matrices whose entries lie in various rings, including the rational numbers, the integers, and finite fields. Most notably, we show that there is no Kochen-Specker coloring of the $n \times n$ idempotent integer matrices for $n \geq 3$, thereby illustrating that Kochen-Specker contextuality is an inherent feature of pure matrix algebra. We apply this to generalize recent no-go results on noncommutative spectrum functors, showing that any contravariant functor from rings to sets (respectively, topologic…
Horn's conjecture, which given the spectra of two Hermitian matrices describes the possible spectra of the sum, was recently settled in the affirmative. In this survey we discuss one of the many steps in this, which required us to introduce a combinatorial gadget called a {\em honeycomb}; the question is then reformulable as about the existence of honeycombs with certain boundary conditions. Another important tool is the connection to the representation theory of the group U(n), by ``classical vs. quantum'' analogies.
A direct sum decomposition theory is developed for direct summands (and complements) of modules over a semiring $R$, having the property that $v+w = 0$ implies $v = 0$ and $w = 0$. Although this never occurs when $R$ is a ring, it always does holds for free modules over the max-plus semiring and related semirings. In such situations, the direct complement is unique, and the decomposition is unique up to refinement. Thus, every finitely generated projective module is a finite direct sum of summands of $R$ (assuming the mild assumption that $1$ is a finite sum of orthogonal primitive idempotents…
We define a formal framework for the study of algebras of type Max-plus, Min-Plus, tropical algebras, and more generally algebras over a commutative idempotent semi-field. This work is motivated by the increasingly diversified use of these algebras which occur also in control theory, automata theory as well as in algebraic geometry, and in more specific ways in other parts of mathematics such as the theory of monoids. In this first article, we expecially re-examine linear algebra over idempotent semi-fields: the most delicate, but undoubtedly the most interesting point is the notion of a singu…
Laws for triangular matrices
该研究探讨域上及含0和1的加法幂等半环上三角矩阵幺半群的恒等式，发现部分此类恒等式无有限公理化的新情况。
In this paper we give a short, elementary proof of a known result in tropical mathematics, by which the convexity of the column span of a zero--diagonal real matrix $A$ is characterized by $A$ being a Kleene star. We give applications to alcoved polytopes, using normal idempotent matrices (which form a subclass of Kleene stars). For a normal matrix we define a norm and show that this is the radius of a hyperplane section of its tropical span.
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