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In this paper, we investigate the interval coupled Sylvester matrix equations which include the well-known (generalized) Sylvester and Lyapunov matrix equations in both real and interval forms. Assuming that certain matrices are simultaneously diagonalizable, we present a fast and efficient approach for enclosing the solution set of the interval coupled Sylvester matrix equations. Our approach, which is a modification of the Krawczyk operator, enables us to reduce the computational complexity considerably. Some numerical tests are given to illustrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Let $ \mathcal {O} $ O denote the ring of integers of a quadratic field $ \mathbb {Q}(\sqrt {-7}) $ Q (− 7). In 2022, Murtuza and Garge [Murtuza N, Garge A. Universality of certain diagonal quadratic forms for matrices over a ring of integers, Indian Journal of Pure and Applied Mathematics, Published online; December 2022.] gave a necessary and sufficient condition for a diagonal quadratic form $ a_1X_1^2+a_2X_2^2+a_3X_3^2 $ a 1 X 1 2 + a 2 X 2 2 + a 3 X 3 2 where $ a_i\in \mathbb {\mathcal {O}} $ a i ∈ O for $ 1\leq i \leq ~3 $ 1 ≤ i ≤ 3 for representing all $ 2\times 2 $ 2 × 2 matrices over $ \mathcal {O} $ O . Let K denote a quadratic field such that its ring of integers $ \mathcal {O}_K $ O K is a principal ideal domain and 2 is a product of two distinct primes. It is a well-known fact that $ \mathbb {Q}(\sqrt {-7}) $ Q (− 7) is the only imaginary quadratic field with the above properties. Let $ D_K $ D K denote the discriminant of K. We have $ D_K\equiv 1(\text{mod }8) $ D K ≡ 1 (mod 8) if and only if 2 is a product of two distinct primes in $ \mathcal {O}_K $ O K . With $ \mathcal {O}_K $ O K as above, in this paper we generalize our earlier result. We give a necessary and sufficient condition for a diagonal quadratic form $ {\sum _{i=1}^{m}a_iX_i^2} $ ∑ i = 1 m a i X i 2 where $ a_i\in \mathcal {O}_K $ a i ∈ O K , $ 1\leq i \leq m $ 1 ≤ i ≤ m to represent all $ 2\times 2 $ 2 × 2 matrices over $ \mathcal {O}_K $ O K . This result is a conjecture stated in [Murtuza N, Garge A. Universality of certain diagonal quadratic forms for matrices over a ring of integers, Indian Journal of Pure and Applied Mathematics, Published online; December 2022]. [ABSTRACT FROM AUTHOR]

A square complex matrix A is EP if it satisfies R (A) = R (A ∗) (or, equivalently, R (A) ⊆ R (A ∗)). This paper aims to introduce two natural generalizations of EP matrices: a square matrix A is defined to be left or right power-EP if it satisfies R (A m) ⊆ R (A ∗) or N (A ∗) ⊆ N (A m) for some integer m ≥ 0 , respectively. Some properties, characterizations and applications of these two classes of matrices are given. [ABSTRACT FROM AUTHOR]

Given a bilinear form on C n , represented by a matrix A ∈ C n × n , the problem of finding the largest dimension of a subspace of C n such that the restriction of A to this subspace is a non-degenerate skew-symmetric bilinear form is equivalent to finding the size of the largest invertible skew-symmetric matrix B such that the equation X ⊤ A X = B is consistent (here X ⊤ denotes the transpose of the matrix X). In this paper, we provide a characterization, by means of a necessary and sufficient condition, for the matrix equation X ⊤ A X = B to be consistent when B is a skew-symmetric matrix. This condition is valid for most matrices A ∈ C n × n . To be precise, the condition depends on the canonical form for congruence (CFC) of the matrix A, which is a direct sum of blocks of three types. The condition is valid for all matrices A except those whose CFC contains blocks, of one of the types, with size smaller than 3. However, we show that the condition is necessary for all matrices A. [ABSTRACT FROM AUTHOR]

In this paper, we consider a linear constraint problem of Hermitian unitary symplectic matrices and its approximation. By constructing a simple unitary matrix U, we verify that Hermitian unitary symplectic matrices are unitary similar to block diagonal Hermitian unitary matrices via U, which simplifies and is crucial to solving the linear constraint problem, and is a special feature of this paper. Then, we solve the linear constraint problem completely, that is deriving the sufficient and necessary conditions of it and inducing Hermitian unitary symplectic solutions to it. We also obtain its optimal approximate solutions. Furthermore, the Procrustes problem of Hermitian unitary symplectic matrices is considered when the linear constraint problem has no solution. [ABSTRACT FROM AUTHOR]

In this paper, we establish a different approach for solving the system of three coupled two-sided Sylvester-type quaternion matrix equations A i X i B i + C i X i + 1 D i = E i , i = 1 , 3 ¯ . We give some new necessary and sufficient conditions for the existence of a solution to this system in terms of Moore-Penrose inverses of the matrices involved. We show that these new solvability conditions are equivalent with the solvability conditions which were presented in a recent paper [Linear Algebra Appl. 2016;496:549–593]. The general solution to the system is given when the solvability conditions are satisfied. Applications that are discussed include the solvability conditions and general η-Hermitian solution to a system of quaternion matrix equations. [ABSTRACT FROM AUTHOR]

Although solvability conditions for a system of two linear equations are well-known even in the case of rings and, for three linear equations, in the case of matrices, in the case of four linear equations there are no results. In this paper, we consider systems of four linear matrix equations A i X B i = C i , i = 1 , 4 ¯ and present some necessary and sufficient conditions for their solvability as well as an expression for the general solution. There are two advantages to our results: the presented solvability conditions in many cases can be presented in a purely algebraic form and the method used in the proof allows for a generalization of the obtained results to some more general structures such as algebras of bounded linear operators or rings, under some additional assumptions concerning regularity only. We present several applications of our results. [ABSTRACT FROM AUTHOR]

The paper considers various problems concerned with the Moore–Penrose inverse of a sum of two matrices. By establishing several original results and by combining various facts known in the literature, the article reveals a number of emerging features of the inverse. The investigations shed also a spotlight on different proving approaches useful to cope with the problems in question. Rather than focusing exclusively on the main topic, the considerations endeavour to place it in a wider context, linking it with different matrix notions, facts, and tools, as well as indicating areas of its applications originating from, e.g. computational methods, physics, statistics, discriminant analysis, or neuroscience. [ABSTRACT FROM AUTHOR]

In this paper, we propose a feasible and effective iteration method for solving the tensor equation A ∗ N X ∗ M B = C with a tensor inequality constraint D ∗ N X ∗ M E ≥ F . The global convergence is established. Numerical examples are provided to illustrate the feasibility and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The finite element model errors mainly come from the complex parts of the geometry, boundary conditions and stress state of the structure. Therefore, the problem for updating mass and stiffness matrices can be reduced to an inverse problem for symmetric matrices with submatrix constraints (IP-MUP): Let Λ = d i a g (λ 1 , ... , λ p) ∈ R p × p and Φ = [ ϕ 1 , ... , ϕ p ] ∈ R n × p be the measured eigenvalue and eigenvector matrices with rank(Φ) = p. Find n × n symmetric matrices M and K such that K Φ = M Φ Λ , Φ ⊤ M Φ = I p , s. t. M (r) = M 0 , K (r) = K 0 , where M(r) and K(r) are the r × r leading principal submatrices of M and K, respectively. We then consider an optimal approximation problem (OAP): Given n × n symmetric matrices Ma and Ka. Find (M ˆ , K ˆ) ∈ S E such that ∥ K ˆ − K a ∥ 2 + ∥ M ˆ − M a ∥ 2 = min (M , K) ∈ S E (∥ K − K a ∥ 2 + ∥ M − M a ∥ 2) , where S E is the solution set of Problem IP-MUP. In this paper, the solvability condition for Problem IP-MUP is established, and the expression of the general solution of Problem IP-MUP is derived. Also, we show that the optimal approximation solution (M ˆ , K ˆ) is unique and derive an explicit formula for it. [ABSTRACT FROM AUTHOR]

This paper introduces and investigates a new class of generalized inverses, called GDMP-inverses (and their duals), as a generalization of DMP-inverses. GDMP-inverses are defined from G-Drazin inverses and the Moore-Penrose inverse of a complex square matrix. In contrast to most other generalized inverses, GDMP-inverses are not only outer inverses but also inner inverses. Characterizations and representations of GDMP-inverses are obtained by means of the core-nilpotent and the Hartwig-Spindelböck decompositions. [ABSTRACT FROM AUTHOR]

The main contribution of this paper is to present new generalized inverses as weaker versions of a G-outer inverse. In particular, we define and characterize left and right G-outer inverses of rectangular matrices. Solvability of matrix equation systems as AXA = AEA and BAEAX = B; or AXA = AEA and XAEAD = D, where A ∈ C m × n , B ∈ C p × m , D ∈ C n × q and E ∈ C n × m , is studied by means of left and right G-outer inverses. The general solution forms of these systems give descriptions of the sets of all left and right G-outer inverses. Using left and right G-outer inverses, we introduce new partial orders and establish their relations with minus partial order and space pre-order. We apply these results to present and investigate left and right G-Drazin inverses of square matrices and corresponding partial orders. [ABSTRACT FROM AUTHOR]

In this paper, we give the complete structures of the equivalence canonical form of four matrices over an arbitrary division ring. As applications, we derive some practical necessary and sufficient conditions for the solvability to some systems of generalized Sylvester matrix equations using the ranks of their coefficient matrices. The results of this paper are new and available over the real number field, the complex number field, and the quaternion algebra. [ABSTRACT FROM PUBLISHER]

This paper is concerned with the conjugate gradient least squares algorithm to solve a class of tensor equations via the Einstein product. The proposed algorithm uses tensor computations with no matricizations involved. We prove that the solution (or the least squares solution) of the tensor equation can be obtained within a finite number of iterative steps in the absence of round-off errors. By selecting the appropriate initial tensor, the least Frobenius norm solution (or the least Frobenius norm least squares solution) of the tensor equation can be obtained. Numerical examples are provided to illustrate the efficiency of the proposed algorithm and testify the conclusions suggested in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the least-squares solution with the least norm to the following system of tensor equations over quaternions A 1 ∗ N X = D 1 , Y ∗ N B 2 = D 2 , A 3 ∗ N X ∗ N B 3 = D 3 , A 4 ∗ N Y ∗ N B 4 = D 4 , A 5 ∗ N X + Y ∗ N B 5 = D 5 , where X , Y are unknown quaternion tensors and the others are given quaternion tensors. Using the expressions of the Moore-Penrose inverses of partitioned tensors, we give a representation of the solution and an algorithm to compute this solution. [ABSTRACT FROM AUTHOR]

In this paper, we obtain new additive perturbation bounds of the Moore-Penrose inverse under the unitarily invariant norm and the Q-norm by using the singular value decomposition, respectively. These bounds always improve the corresponding ones in [Cai L et al. Additive and multiplicative perturbation bounds for the Moore-Penrose inverse. Linear Algebra Appl. 2011;434:480–489]. [ABSTRACT FROM AUTHOR]

As is known, Alternating-Directional Doubling Algorithm (ADDA) is quadratically convergent for computing the minimal nonnegative solution of an irreducible singular M-matrix algebraic Riccati equation (MARE) in the noncritical case or a nonsingular MARE, but ADDA is only linearly convergent in the critical case. The drawback can be overcome by deflating techniques for an irreducible singular MARE so that the speed of quadratic convergence is still preserved in the critical case and accelerated in the noncritical case. In this paper, we proposed an improved deflating technique to accelerate further the convergence speed – the double deflating technique for an irreducible singular MARE in the critical case. We proved that ADDA is quadratically convergent instead of linearly when it is applied to the deflated algebraic Riccati equation (ARE) obtained by a double deflating technique. We also showed that the double deflating technique is better than the deflating technique from the perspective of dimension of the deflated ARE. Numerical experiments are provided to illustrate that our double deflating technique is effective. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear matrix equation X p = A # t (M T (X − 1 + B) − 1 M) , where p ≥ 1 is a positive integer, t ∈ [ 0 , 1 ] , M is an n × n nonsingular matrix, A is a positive definite matrix and B is a positive semidefinite matrix, is considered. The notation A # t B is the t-weighted geometric mean of the positive definite matrices A and B. Based on the properties of the Thompson metric, we prove that the nonlinear matrix equation always has a unique positive definite solution and we compare it with the unique positive definite solution of the equation X p = A + M T (X − 1 + B) − 1 M , which has been studied in Jung et al. [On the solution of the nonlinear matrix equation X n = f (X). Linear Algebra Appl. 2009;430:2042–2052]; Meng and Kim [The positive definite solution of the nonlinear matrix equation X p = A + M (B + X − 1 ) − 1 M ∗ . J Comput Appl Math. 2017;322:139–147]. A fixed-point iteration method for obtaining the unique positive definite solution and an elegant estimate of the solution are given. Perturbation analysis of the unique positive definite solution is presented. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the m -core inverse in the Minkowski space, and get a sufficient and necessary condition for the existence of the inverse and some other related properties. Furthermore, by using the inverse, we introduce the m -core partial ordering and obtain solutions (or restricted least squares solutions) of some matrix equations in the Minkowski space. [ABSTRACT FROM AUTHOR]

In this paper, necessary and sufficient conditions for the existence of the Hermitian positive definite solution for a nonlinear matrix equation are derived. Then, a sufficient condition for the existence of the unique Hermitian positive definite solution is presented. Further, we propose to use Newton's method to solve this nonlinear matrix equation with some constraints. In addition, we prove the convergence of Newton's method. Finally, we present some numerical examples to illustrate the effectiveness of the derived results. [ABSTRACT FROM AUTHOR]

Letbe the set of allmatrices over the real quaternion algebra.,,, whereis the conjugate of the quaternion. We call thatis-Hermitian, if,;is-bihermitian, if. We in this paper, present the solvability conditions and the general-Hermitian solution to a system of linear real quaternion matrix equations. As an application, we give the necessary and sufficient conditions for the systemto have an-bihermitian solution. We establish an expression of the-bihermitian to the system when it is solvable. We also obtain a criterion for a quaternion matrix to be-bihermitian. Moreover, we provide an algorithm and a numerical example to illustrate the theory developed in this paper. [ABSTRACT FROM PUBLISHER]

This paper investigates the problem of establishing exact formulas for calculating the maximum and minimum ranks of matrix expressions or matrix equalities that involve two matrices and their generalized inverses of matrices. For a pair of matricesAandBsuch that the productABis defined, two fundamental matrix expressions or matrix equalities composed by their generalized inverses are given byand, whereandare the-inverses ofAandB, respectively. Recently, the present author established a group of exact formulas for calculating the max–min ranks of the products, and derived many fundamental algebraic properties offrom the rank formulas. As a continuation, we study in this paper the ranks of the multiple matrix products,,,,, as well as the matrix differences,,,. All these kinds of problems are in fact to characterize algebraic properties of certain matrix-valued functions and we need to utilize a variety of known matrix rank formulas during this tedious work. [ABSTRACT FROM PUBLISHER]

One of the fundamental research problems in the theory of generalized inverses has certainly been establishments of various matrix equalities that involve generalized inverses. A simplest form of such matrix equalities is given by A (i , ... , j) = B (k , ... , l) , where A and B are matrices of the same size, and (⋅) (i , ... , j) denotes the { i , ... , j } -generalized inverse of a matrix. Because generalized inverses of a matrix are not necessarily unique, the equality A (i , ... , j) = B (k , ... , l) does not imply A=B for singular matrices. In this paper, we derive necessary and sufficient conditions for this kind of equalities to hold for the eight commonly-used types of generalized inverse of A and B using the matrix equation method and the matrix rank method. [ABSTRACT FROM AUTHOR]

In this paper, the solvability conditions and the explicit expressions of the Hermitian solutions to the system of matrix equations and the Hermitian nonnegative definite solutions to the system of matrix equations are, respectively, put forward, by making full use of the generalized inverse and the rank of matrices. As applications, some special cases of the above systems of matrix equations are considered. In addition, the maximal ranks and inertias of the Hermitian solutions are, respectively, presented. [ABSTRACT FROM PUBLISHER]

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]

In this paper, we study a particular class of matrices generated by generalized permutation matrices corresponding to a subgroup of some permutation group. As applications, we first present a technique from which we can get closed formulas for the roots of many families of polynomial equations with degree between 5 and 10, inclusive. Then, we describe a tool that shows how to find solutions to Fermat's last theorem and Beal's conjecture over the square integer matrices of any dimension. Finally, simple generalizations of some of the concepts in number theory to integer square matrices are presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider an expression of the general solution to the classical system of matrix equationsWe present a necessary and sufficient condition for the existence of a solution to the system by using generalized inverses. We give an expression of the general solution to the system when it is solvable. As applications, we derive some necessary and sufficient conditions for the consistence to the systemand the systemwheremeans conjugate transpose. We also give the expressions of the general solutions to the systems. [ABSTRACT FROM AUTHOR]

In this paper,we discuss rank-constrained least squares solutions to the matrix equationunder the rank restrictionin the Frobenius norm. We derive the rank range and expression of these least squares solutions by applying generalized inverses, singular value decomposition and the Eckart–Young–Mirsky theorem. [ABSTRACT FROM AUTHOR]

In this paper, we use the well-known Löwner order and the core partial order to introduce a new partial orderon the class of core matrices which is not dominated by any of the known matrix partial orders. We characterize, study its relations with the Löwner partial order under constraints, and exemplify its differences with other partial orders. [ABSTRACT FROM PUBLISHER]

The combined matrix of a nonsingular matrixAis the Hadamard (entrywise) product. This paper deals with the characterization of the diagonal entries of a combined matrixC(A) of a given nonsingular real matrixA. A partial answer describing the diagonal entries ofC(A) in the positive definite case was given by Fiedler in 1964. Recently in 2011, Fiedler and Markham characterized the sequence of diagonal entries of the combined matrixC(A) for any totally positive matrixAwhen the size is 3. For this case, we characterize totally negative matrices and we find necessary and sufficient conditions for the sequence of diagonal entries ofC(A), in both cases, symmetric and nonsymmetric. [ABSTRACT FROM PUBLISHER]

Assume that a pair of general Linear Random-effects Models (LRMs) are given with a correlated covariance matrix for their error terms. This paper presents an algebraic approach to the statistical analysis and inference of the two correlated LRMs using some state-of-the-art formulas in linear algebra and matrix theory. It is shown first that the best linear unbiased predictors (BLUPs) of all unknown parameters under LRMs can be determined by certain linear matrix equations, and thus the BLUPs under the two LRMs can be obtained in exact algebraic expressions. We also discuss algebraical and statistical properties of the BLUPs, as well as some additive decompositions of the BLUPs. In particular, we present necessary and sufficient conditions for the separated and simultaneous BLUPs to be equivalent. The whole work provides direct access to a very simple algebraic treatment of predictors/estimators under two LRMs with correlated covariance matrices. [ABSTRACT FROM PUBLISHER]

In this paper, we study the reverse order law for the least square-inverse of an operator productusing the technique of matrix form of bounded linear operators. In particular, some necessary and sufficient conditions for theinclusionsandare presented. Moreover, some finite dimensional results are extended to infinite dimensional settings. [ABSTRACT FROM PUBLISHER]

This paper focuses on Bezout equations derived from multivariate polynomial matrices in which relationships between one primary variable and other variables are described by real entire functions. We propose a method for obtaining a solution belonging to a set of multivariate rational function matrices in which all entries are real entire functions with respect to the primary variable. The proposed method is based on a new approach that overcomes the constraints and difficulties due to many variables by expanding a class of solutions to multivariate rational function matrices. [ABSTRACT FROM PUBLISHER]

In this paper, we derive an interesting result that the mixed-type reverse order lawshold if and only if the matrix productis invariant, whereis taken, respectively, asas well as. [ABSTRACT FROM AUTHOR]

This study addresses Bezout equations over bivariate polynomial matrices, where the relationship between two variables is described by a real entire function. This paper proposes a solution method that makes optimal use of minor primeness to reduce such Bezout equations to simple equations over univariate scalar polynomials. The proposed solution method requires only matrix calculations, thus making it very useful, especially in the absence of modern computer algebra systems. [ABSTRACT FROM AUTHOR]

Letandbe two real symmetric orthogonal matrices. Anreal matrixis said to be an-symmetric matrix if. The-symmetric matrices have been widely used in engineering and scientific computing. In this paper, we construct an algorithm to solve the general coupled matrix equations, whereis a-symmetric matrix. The algorithm produces suitablesuch thatis minimized within a finite number of iteration steps in the absence of round-off errors. We show that the algorithm is stable any case. The algorithm requires little storage capacity. Numerical examples are given to show that the algorithm is efficient. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following minimization problem:where,,,andare given. An efficient inequality relaxation technique is presented to relax the matrix inequality constraint so that there is an optimal solution which is(R,S)-symmetric that minimize, and also satisfies the corrected matrix inequality constraint. A hybrid algorithm with convergence analysis is given to solve this problem. Numerical examples show that the algorithm requires less CPU times when compared with some other methods. [ABSTRACT FROM AUTHOR]

This paper proposes a reduction technique for the generalized Riccati difference equation arising in optimal control and optimal filtering. This technique relies on a study on the generalized discrete algebraic Riccati equation. In particular, an analysis on the eigenstructure of the corresponding extended symplectic pencil enables to identify a subspace in which all the solutions of the generalized discrete algebraic Riccati equation are coincident. This subspace is the key to derive a decomposition technique for the generalized Riccati difference equation. This decomposition isolates a “nilpotent” part, which converges to a steady-state solution in a finite number of steps, from another part that can be computed by iterating a reduced-order generalized Riccati difference equation. [ABSTRACT FROM PUBLISHER]

Every associative algebra has an associated Lie algebra. For a matrix algebra, there is a linear transformation associated with this Lie product by fixing one variable. The well-known dimensional formula of the kernel is due to Frobenius. Subsequently, Gracia obtained the dimensional formulas of the kernels of the second and third powers of the transformation. We fix two matrices and obtain a linear transformation. By using techniques from elementary linear algebra, together with the image spaces of the powers of the transformation, this paper provides an alternative approach to this problem. We obtain the dimensional formulas for kernels of each power of the transformation. Furthermore, the basis for kernels of powers of the transformation is described explicitly. In particular, we give the general solution forfor thek-fold Lie product with. [ABSTRACT FROM PUBLISHER]

We find an algorithmic procedure that enables the computation and description of the structure of the isotropy subgroups of the group of complex orthogonal matrices with respect to the action of similarity on complex symmetric matrices. A key step in our proof is to solve a certain rectangular block upper triangular Toeplitz matrix equation. [ABSTRACT FROM AUTHOR]

We establish several necessary and sufficient conditions for the existence and uniqueness of Hermitian positive definite (HPD) solutions to the general matrix equation X p = A + ∑ i = 1 m M i ∗ (B + X − 1 ) − 1 M i , where p, m are positive integers, Mi (i = 1, 2, ..., m) are n × n nonsingular complex matrices, A and B are HPD matrices, and then give three algorithms to get the unique solution. Moreover, we give two numerical examples to illustrate the effectiveness of the theoretical results and the behavior of the considered methods. [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]

Let UTm be the algebra of all m × m upper triangular matrices over a field F whose characteristic is different from 2. Given an involution * of the first kind on UTm, we will obtain the minimal integer t such that the Lie polynomial [[z1, z2], ..., [z2t−1, z2t]] is an identity for K (m , ∗) = { U ∈ U T m ∣ U ∗ = − U }. Afterwards, under a mild technical restriction on F , we will describe all multilinear Lie polynomials whose image evaluated on K(m, *) is the space formed by all strictly upper triangular matrices of K(m, *). [ABSTRACT FROM AUTHOR]

Necessary and sufficient conditions are obtained for a pair of matrix equations AX = C and XB = D to have common Re-nnd and Re-pd solutions and the explicit representations of the general common Re-nnd and Re-pd solutions are given when the stated conditions hold. [ABSTRACT FROM AUTHOR]

We propose an algorithm for finding the so-called principal solution of the Sylvester matrix equation over max-plus algebra. The derivation of our algorithm is based on the concept of tropical tensor product introduced by Butkovič and Fiedler. Our algorithm reduces the computational cost of finding the principal solution from quartic to cubic. It also reduces the space complexity from quartic to quadratic. Since matrix–matrix multiplication is the most important ingredient of our proposed technique, we show how to use column-oriented matrix multiplications in order to speed-up MATLAB implementation of our algorithm. Finally, we illustrate our results and discuss the connection with the residuation theory. [ABSTRACT FROM AUTHOR]

In this work, we study the parametric Sylvester matrix equation A (p) X + X B (p) = C (p) whose elements are linear functions of uncertain parameters varying within some intervals. We first explore some properties of its solution set and then present some sufficient conditions for boundedness of the solution set. The main work is developing a modified variant of the parametric Krawczyk operator which reduces the computational complexity of enclosing the solution set significantly, compared to the parametric Krawczyk operator on the Kronecker product form. Some numerical experiments are given to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We present some closed form expressions for the Cauchy–Sylvester compound determinant and the determinant of the Adjugate matrix when each minor in their expressions is replaced by some linear functions (including the sum) and products of its elements. [ABSTRACT FROM AUTHOR]

Let R be an exchange ring. We prove that the relative elementary subgroups E n (R , I) are normal in the general linear group G L n (R) if n ≥ 1 and that the standard commutator formula E n (R , I) = [ E n (R) , E n (R , I) ] = [ E n (R) , C n (R , I) ] holds if n ≥ 3. Moreover, we classify the subgroups of G L n (R) that are normalized by the elementary subgroup E n (R) in the case n ≥ 3. [ABSTRACT FROM AUTHOR]

The following second-order conjugate matrix equation is considered X H A X + B H X + X H B + D = 0 , where A H = A ∈ C n × n , B ∈ C n × p , D H = D ∈ C p × p are known matrices and X ∈ C n × p is an unknown matrix to be determined. The necessary and sufficient condition for the matrix equation to have a solution is established and the explicit solutions of the equation are provided. [ABSTRACT FROM AUTHOR]

Within the framework of the theory of quaternion row-column determinants previously introduced by the author, we derive determinantal representations (analogs of Cramer's rule) of a solution to the system of two-sided quaternion matrix equations A 1 X B 1 = C 1 , A 2 X B 2 = C 2 and its special cases with one one-sided equation when B 1 = I or A 1 = I , where I is an identity matrix of appropriate shape. Since the Moore–Penrose inverse is a necessary tool to solve matrix equations, we use determinantal representations of the Moore–Penrose inverse previously obtained by the theory of row-column determinants as well. [ABSTRACT FROM AUTHOR]