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This paper considers the problem of testing statistical hypotheses about the variance components under linear models with crossed random effects. The objective here is two-fold. First, new derivations of exactly distributed F test statistics are presented. Second, an alternative Monte Carlo permutation procedure is proposed to approximate the distribution of the F statistics, which shows its usefulness when the error components distributions depart from normality. Transformations that uniquely decompose the covariance structure of the response vector and do not sacrifice any part of the data, as existing methods do, are presented in matrix form. The suggested transformations highlight the exchangeable covariance structure of the model under the null hypotheses of interest and thus motivate the use of the permutation procedure. Comments on the performance of the proposed procedure compared to existing tests as well as using a real data example are provided. [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.

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]

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.

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 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.

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]

This article brings together miscellaneous formulas and facts on matrix expressions that are composed by idempotent matrices in one place with cogent introduction and references for further study. The author will present the basic mathematical ideas and methodologies of the matrix analytic theory in a readable, up-to-date, and comprehensive manner, including constructions of various algebraic matrix identities composed by the conventional operations of idempotent matrices, and uses of the block matrix method in the derivation of closed-form formulas for calculating the ranks of matrix expressions that are composed by idempotent matrices. The author also determines the maximum and minimum ranks of some matrix pencils composed by the products of matrices and their generalized inverses and uses the ranks to characterize algebraic performance of the matrix pencils.

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.)

Reverse order laws for generalized inverses of matrix products are a classic object of study in the theory of generalized inverses. One of the well-known reverse order laws for a matrix product AB is (AB)(i,…,j)=B(s2,…,t2)A(s1,…,t1), where (⋅)(i,…,j) denotes a {i,…,j}-generalized inverse of matrix. Because {i,…,j}-generalized inverse of a singular matrix is not unique, the relationships between both sides of the reverse order law can be divided into four situations for consideration. The aim of this paper is to give an overview of plenty of results concerning reverse order laws for {i,…,j}-generalized inverses of the product AB, from the development of background and preliminary tools to the collection of miscellaneous formulas and facts on the reverse order laws in one place with cogent introduction and references for further study. We begin with the introduction of a linear mixed model y=ABβ+Aγ+ϵ and the presentation of two least-squares methodologies for estimating the fixed parameter vector β in the model, and the description of connections between the two types of least-squares estimators and the reverse order laws for generalized inverses of AB. We then prepare various necessary matrix study tools, including a general theory on linear or nonlinear algebraic matrix identities, a group of expansion formulas for calculating ranks of block matrices, two groups of explicit formulas for calculating the maximum and minimum ranks of B(s2,…,t2)A(s1,…,t1), as well as necessary and sufficient conditions for B(s2,…,t2)A(s1,…,t1) to be invariant with respect to the choice of the generalized inverses, etc. Subsequently, we present a unified approach to the 512 matrix set inclusion problems associated with the above reverse order laws for the eight commonly-used types of generalized inverses of A, B, and AB by means of the definitions of generalized inverses, the block matrix methodology (BMM), the matrix equation methodology (MEM), and the matrix rank methodology (MRM).

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]

We give the quasi-Euclidean classification of the maximal (with respect to the f-vector) alcoved polyhedra. The f-vector of these maximal convex bodies is (20 , 30 , 12) , so they are simple dodecahedra. We find eight quasi-Euclidean classes. This classification, which preserves angles, is finer than the known combinatorial classification (found in 2012 by Jiménez and de la Puente), which has only six classes. Each alcoved polyhedron P is represented by a unique visualized idempotent matrix A. Some 2-minors of A are invariants of P : they are the tropical edge-lengths of P . [ABSTRACT FROM AUTHOR]

A square matrix is said to be diagonalizable if it is similar to a diagonal matrix. In this paper, we discuss diagonability of matrices over commutative semirings and give an equivalent condition for an idempotent matrix over a commutative semiring to be diagonalizable. Also, we obtain an equivalent description for a matrix over a multiplicatively cancellative and commutative semiring to be diagonalizable. [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]

The equality of rank and trace for an idempotent matrix with entries from an arbitrary field is established, using a linear transformation approach. This proof makes use of the notion of a basis without referring to matrix decomposition so that it is substantially simpler than previous proofs found in the literature. [ABSTRACT FROM AUTHOR]

In this work, we are interested in the subgraph of unitary Cayley graph of a matrix algebra over a finite field induced by the set of idempotent matrices. We study its structure and completely obtain all connected components with the degree on each vertex. Most of them turn out to be a regular bipartite graph such that each partite set has the same number of vertices. Our main tools are combinatorial results on matrix algebras. Moreover, we compute the clique number, chromatic number and independence number of the graph. [ABSTRACT FROM AUTHOR]

For an n -by- n matrix A , we determine when the numerical ranges W (A k) , k ≥ 1 , of powers of A are all equal to each other. More precisely, we show that this is the case if and only if A is unitarily similar to a direct sum B ⊕ C , where B is idempotent and C satisfies W (C k) ⊆ W (B) for all k ≥ 1. We then consider, for each n ≥ 1 , the smallest integer k n for which every n -by- n matrix A with W (A) = W (A k) for all k , 1 ≤ k ≤ k n , has an idempotent direct summand. For each n ≥ 1 , let p n be the largest prime less than or equal to n + 1. We show that (1) k n ≥ p n for all n , (2) if A is normal of size n , then W (A) = W (A k) for all k , 1 ≤ k ≤ p n , implies A having an idempotent summand, and (3) k 1 = 2 and k 2 = k 3 = 3. These lead us to ask whether k n = p n holds for all n ≥ 1. [ABSTRACT FROM AUTHOR]

In the present paper, we study diagonability of matrices over a duo ring. We prove that an idempotent matrix over such a ring is diagonalizable under a similarity transformation. [ABSTRACT FROM AUTHOR]

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 if the transpose of every 2 × 2 {2\times 2} idempotent matrix over a division ring D, different from the identity matrix, is not invertible, then D is commutative. [ABSTRACT FROM AUTHOR]

We introduce and study a congruence ρ on the normal tropical matrix semiring M n N , which is relevant to the Kleene stars of normal tropical matrices. We prove that this congruence is a bisemilattice congruence and give an exact description of each ρ -class. In particular, we show that the ρ -class Eρ is an interval when E is a strongly regular normal tropical matrix. We also present a method using Floyd–Warshall algorithm to compute the greatest lower bound of Eρ in such case. [ABSTRACT FROM AUTHOR]

The main goal of this paper is the numerical computation of solutions of the so-called Yang–Baxter-like matrix equation A X A = X A X , where A is a given complex square matrix. Two novel matrix iterations are proposed, both having second-order convergence. A sign modification in one of the iterations gives rise to a third matrix iteration. Strategies for finding starting approximations are discussed as well as a technique for estimating the relative error. One of the methods involves a very small cost per iteration and is shown to be stable. Numerical experiments are carried out to illustrate the effectiveness of the new methods. [ABSTRACT FROM AUTHOR]

Let R be any ring with identity. An element a ∊ R is called nil-clean, if a = e + n where e is an idempotent element and n is a nil-potent element. In this paper we give necessary and sufficient conditions for a 2 × 2 matrix over an integral domain R to be nil-clean. [ABSTRACT FROM AUTHOR]

Let A be a matrix with A − 1 = A . We find all commuting solutions and some non-commuting solutions of the matrix equation A X A = X A X . All non-commuting solutions are also obtained under a full rank condition. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Central China Normal University is the property of Huazhong Normal 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.)

In this paper, we have established the existence and representation of group inverse for 2×2 block matrix over the field of complex numbers C inMinkowski spaceM. Further, for various types of block matrices the characterization theorems for the existence of group inverse are determined. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Central China Normal University is the property of Huazhong Normal 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.)

This paper contains the algebraic analog of universal classifying bundles and Chern classes. We imitate the topological counterpart of universal bundles over the Grassmannian to construct some graded commutative differential algebras O *(K [X]/(X2 - X, trX - r)) and O*(K [X]/(X² - X)), whose corresponding cohomology are polynomial algebras isomorphic to K[c1, . . ., cr] and K[c1, c2, . . .] respectively, for the Chern classes cp with p ≥ 1, for the field K = Q, R or C. Here X denotes the infinite matrix X = [Xpq], Xn denotes the corresponding matrix obtained from X by setting to zero the entries Xpq when p > n or q > n, and (X² - X, trX - r) (resp. (X² - X)) denotes the ideal generated by the power series ∑ p Xpp - r and the entries of the matrix X² - X (resp. the entries of X² - X). [ABSTRACT FROM AUTHOR]

In a recent paper C. Miguel proved that the diameter of the commuting graph of the matrix ring Mn(ℝ) is equal to 4 if either n = 3 or n ≥ 5. But the case n = 4 remained open, since the diameter could be 4 or 5. In this work we close the problem showing that also in this case the diameter is 4. [ABSTRACT FROM AUTHOR]

Let A be an idempotent matrix. We obtain an explicit expression for all the solutions of the quadratic matrix equation A X A = X A X , completing the task of finding general solutions of the equation explicitly with a given idempotent matrix A . [ABSTRACT FROM AUTHOR]

Letandbe two nonzero commutingand-quadratic matrices, respectively, wherewithand. The aim of this work is mainly to characterize all situations, where the linear combinationis a generalized quadratic matrix. The results established here cover many of the results in the literature related to idempotency, involutivity and tripotency of the linear combinations of idempotent and/or involutive matrices. [ABSTRACT FROM AUTHOR]

This paper presents necessary and sufficient conditions for a square matrix over an arbitrary field to be expressed as a product of idempotent and square-zero matrices. Since the similarity structures of such factors are completely determined by their rank, we also investigate the ranks that these factors may have. The case where the factors are all idempotent or all square-zero have been investigated by various authors, some of whom are listed in the references of this paper. [ABSTRACT FROM AUTHOR]

An element of a ring R is called nil-clean if it is the sum of an idempotent and a nilpotent element. A ring is called nil-clean if each of its elements is nil-clean. S. Breaz et al. in [1] proved their main result that the matrix ring Mn(F) over a field F is nil-clean if and only if F ≅ F2, where F2 is the field of two elements. M. T. Kosan et al. generalized this result to a division ring. In this paper, we show that the n x n matrix ring over a principal ideal domain R is a nil-clean ring if and only if R is isomorphic to F2. Also, we show that the same result is true for the 2x2 matrix ring over an integral domain R. As a consequence, we show that for a commutative ring R, if M2(R) is a nil-clean ring, then dimR = 0 and charR/J(R) = 2. [ABSTRACT FROM AUTHOR]

In this paper, we construct all classes of 3×3 strongly clean upper triangular matrices over Z. Moreover, necessary and sufficient conditions under which a 3×3 upper triangular matrix over Z is strongly clean are given. [ABSTRACT FROM AUTHOR]

It is investigated the necessary and sufficient conditions for the generalized quadraticity of a linear combination of any two generalized quadratic matrices. The main result obtained is, in a sense, a generalization of the main results given in [Uç M, Özdemir H, Özban AY. On the quadraticity of linear combinations of quadratic matrices. Linear Multilinear Algebra. 2015;63:1125–1137.] which contains many of the results in the literature related to idempotency or involutivity of the linear combinations of idempotent and/or involutive matrices, to the generalized quadratic matrices. [ABSTRACT FROM AUTHOR]

This paper is a correction paper concerned with an important theorem about an involution with Engel condition. [ABSTRACT FROM PUBLISHER]