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This paper establishes the solvability conditions and an expression of the exact solution to a system of three Sylvester-like quaternion tensor equations in four variables. Based on a comprehensive analysis of the general solution and the solvability conditions associated with the system, necessary and sufficient conditions are deduced to a system of Sylvester-like tensor equations, including the unknowns as η-Hermitian quaternion tensors. Ultimately, we design an algorithm to compute the general solution, even a numerical example to illustrate the essential findings of this paper. [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]

In this paper, we deal with the construction of symmetric matrix whose corresponding graph is connected and unicyclic using some pre-assigned spectral data. Spectral data for the problem consist of the smallest and the largest eigenvalues of each leading principal submatrices. Inverse eigenvalue problem (IEP) with this set of spectral data is generally known as the extremal IEP. We use a standard scheme of labeling the vertices of the graph, which helps in getting a simple relation between the characteristic polynomials of each leading principal submatrix. Sufficient condition for the existence of the solution is obtained. The proof is constructive, hence provides an algorithmic procedure for finding the required matrix. Furthermore, we provide the condition under which the same problem is solvable when two particular entries of the required matrix satisfy a linear relation. [ABSTRACT FROM AUTHOR]

In this paper, we study the Gauss-Seidel and successive over-relaxation iteration methods for quaternion linear systems A x = b and obtain the structure-preserving algorithms of Gauss-Seidel and successive over-relaxation iteration methods for quaternion linear systems A x = b . The convergence and computational cost of these iteration methods are discussed. Numerical examples are given to demonstrate the efficiency of structure-preserving algorithms of Gauss-Seidel iteration and successive over-relaxation iteration methods. As an application, we apply two kinds of structure-preserving iterative algorithms to solve elliptic biquaternion linear systems A x = b . [ABSTRACT FROM AUTHOR]

In this paper, we establish some necessary and sufficient conditions for the solvability to a system of five quaternion matrix equations in terms of the Moore–Penrose inverse and the rank of a matrix, and give an expression of the general solution to the system when it is consistent. As an application, we investigate an η -Hermicity solution of a system. Moreover, we present a numerical example to illustrate the main results of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider skew-Hermitian solution of coupled generalized Sylvester matrix equations encompassing ∗ -hermicity over complex field. The compact formula of the general solution of this system is presented in terms of generalized inverses when some necessary and sufficient conditions are fulfilled. An algorithm and a numerical example are provided to validate our findings. A numerical example is carried out using determinantal representations of the Moore–Penrose inverse. [ABSTRACT FROM AUTHOR]

This paper introduces a new class of generalized inverses for square matrices: core-EP G-Drazin (CEPGD) inverse. The CEPGD inverse is not unique and defined as a proper composition of the core-EP and the G-Drazin inverse. Representations of CEPGD inverses related to the core-nilpotent decomposition and the Hartwig–Spindelböck decomposition are established. The existence of CEPGD inverses as well as a few characterizations and representations of this inverse are discussed. In addition, we consider some additional properties of the CEPGD inverses through an induced binary relation. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the Moore–Penrose inverse of a square matrix based on the canonical polar decomposition by using an invertible matrix and a partial isometry matrix, and give a characterization of the B-T inverse by using matrix equations. Furthermore, we introduce a generalized group inverse (we call it the ℋ-group inverse), and get some properties and characterizations of the inverse. [ABSTRACT FROM AUTHOR]

Purpose: In this paper, the authors study the nonlinear matrix equation Xp=Q±A(X-1+B)-1AT, that occurs in many applications such as in filtering, network systems, optimal control and control theory. Design/methodology/approach: The authors present some theoretical results for the existence of the solution of this nonlinear matrix equation. Then the authors propose two iterative schemes without inversion to find the solution to the nonlinear matrix equation based on Newton's method and fixed-point iteration. Also the authors show that the proposed iterative schemes converge to the solution of the nonlinear matrix equation, under situations. Findings: The efficiency indices of the proposed schemes are presented, and since the initial guesses of the proposed iterative schemes have a high cost, the authors reduce their cost by changing them. Therefore, compared to the previous scheme, the proposed schemes have superior efficiency indices. Originality/value: Finally, the accuracy and effectiveness of the proposed schemes in comparison to an existing scheme are demonstrated by various numerical examples. Moreover, as an application, by using the proposed schemes, the authors can get the optimal controller state feedback of $x(t+1) = A x(t) + C v(t)$. [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]

This paper mainly gives some sufficient and necessary conditions for an element in a ring to be EP , partial isometry and strongly EP by some equalities, using the solutions of certain equations and constructing the invertible elements in a ring. [ABSTRACT FROM AUTHOR]

This paper studies a system of three Sylvester-type quaternion matrix equations with ten variables A i X i + Y i B i + C i Z i D i + F i Z i + 1 G i = E i , i = 1 , 3 ¯ . We derive some necessary and sufficient conditions for the existence of a solution to this system in terms of ranks and Moore-Penrose inverses of the matrices involved. We present the general solution to the system when the solvability conditions are satisfied. As applications of this system, we provide some solvability conditions and general solutions to some systems of quaternion matrix equations involving ϕ-Hermicity. Moreover, we give some numerical examples to illustrate our results. The findings of this paper extend some known results in the literature. [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 derive a Toeplitz-structured closed form of the unique positive semi-definite stabilizing solution for the discrete-time algebraic Riccati equations, especially for the case that the state matrix is not stable. Based on the found form and fast Fourier transform, we propose a new algorithm for solving both discrete-time and continuous-time large-scale algebraic Riccati equations with low-rank structure. It works without unnecessary assumptions, complicated shift selection strategies, or matrix calculations of the cubic order with respect to the problem scale. Numerical examples are given to illustrate its features. Besides, we show that it is theoretically equivalent to several algorithms existing in the literature in the sense that they all produce the same sequence under the same parameter setting. [ABSTRACT FROM AUTHOR]

In this paper we consider linear gyroscopic mechanical systems.More precisely, we consider the perturbation theory for stable gyroscopic systems. We present new relative perturbation bounds for the eigenvalues as well as the bounds for the perturbation of the corresponding eigenspaces. Derived bounds are dependent only on system matrices of the original and perturbed systems. The quality of obtained results is illustrated in the numerical experiment. [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]

In this paper, we establish some efficient iteration methods for solving the general coupled discrete-time periodic matrix equation. More concretely, we propose some structure preserving matrix versions of subspace methods, such as bi-conjugate gradient, bi-conjugate residual, conjugate gradient squared, bi-conjugate gradient stabilized, generalized minimal residual and restarted generalized minimal residual methods. We demonstrate experimentally that the proposed iteration methods are feasible and efficient for the periodic matrix equation. [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]

Following the popularity of the core-EP (c-EP) and weighted core-EP (w-c-EP) inverses, so called one-sided versions of the w-c-EP inverse are introduced recently in Behera et al. (Results Math 75:174 (2020). These extensions are termed as E-w-c-EP and F-w-d-c-EP g-inverses as well as the star E-w-c-EP and the F-w-d-c-EP star classes of g-inverses. The applicability of these g-inverses in solving certain restricted matrix equations has been verified. Several additional results on these classes of g-inverses are established in this paper. In addition, the Moore–Penrose E-w-c-EP inverse and the F-w-d-c-EP Moore–Penrose inverse are proposed using proper expressions that involve the Moore–Penrose inverse and the E-w-c-EP or F-we-d-c-EP inverse. Further, the W-weighted Moore–Penrose c-EP and the W-weighted c-EP Moore–Penrose g-inverses are considered with the aim to extend the considered w-c-EP generalized inverses to rectangular matrices. Characterizations, properties, representations and applications of these inverses are considered. [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 propose a new inversion-free iterative method for computation of positive definite solution of the nonlinear matrix equation X p = A + M (B + X - 1 ) - 1 M ∗ , where p ≥ 1 is a positive integer, A and B are Hermitian positive semidefinite matrices, and M is an arbitrary square complex matrix. This matrix equation has been studied recently in Meng and Kim (J Compt Appl Math 322:139–147, 2017), where the authors proposed an inversion-free algorithm for solving this equation with the hypothesis that the matrix B is nonsingular. For our part, we propose a new algorithm that is applicable for all choices of the positive semidefinite matrix B even if it is singular. To prove the convergence of the proposed algorithm, we prove a new matrix inequality. The efficiency of the proposed algorithm is confirmed by some numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the m -core-EP inverse in Minkowski space, consider its properties, and get several sufficient and necessary conditions for the existence of the m -core-EP inverse. We give the m -core-EP decomposition in Minkowski space, and note that not every square matrix has the decomposition. Furthermore, by applying the m -core-EP inverse and the m -core-EP decomposition, we introduce the m -core-EP order and give some characterizations of it. [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, some accelerated iterative algorithms are developed to find the minimal positive solution of the nonsymmetric algebraic Riccati equation arising in transport theory. The convergence analysis shows that the sequences of vectors generated by iterative algorithms with the initial vector (e , e) are monotonically increasing and converge to the minimal positive solution of the vector equations. Numerical examples are provided to illustrate the efficiency of the proposed algorithms and testify the conclusions suggested in this paper. [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]

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]

The solution of a large-scale Sylvester matrix equation plays an important role in control and large scientific computations. In this paper, we are interested in the large Sylvester matrix equation with large dimensionA and small dimension B, and a popular approach is to use the global Krylov subspace method. In this paper, we propose three new algorithms for this problem. We first consider the global GMRES algorithm with weighting strategy, which can be viewed as a precondition method. We present three new schemes to update the weighting matrix during iterations. Due to the growth of memory requirements and computational cost, it is necessary to restart the algorithm effectively. The deflation strategy is efficient for the solution of large linear systems and large eigenvalue problems; to the best of our knowledge, little work is done on applying deflation to the (weighted) global GMRES algorithm for large Sylvester matrix equations. We then consider how to combine the weighting strategy with deflated restarting, and propose a weighted global GMRES algorithm with deflation for solving large Sylvester matrix equations. In particular, we are interested in the global GMRES algorithm with deflation, which can be viewed as a special case when the weighted matrix is chosen as the identity. Theoretical analysis is given to show rationality of the new algorithms. Numerical experiments illustrate the numerical behavior of the proposed algorithms. [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]

This paper is concerned with the problem of the optimal approximation for a given matrix pencil (Ma, Da, Ga, Ka, Na) under the spectral constraint and the symmetric constraint. Such a problem arises in finite element model updating for damped gyroscopic systems. By using constrained optimization theory and matrix derivatives, an explicit formulation for the solution of the problem is established. The efficiency and accuracy of the proposed method is numerically verified by a simple five-degree-of-freedom system. [ABSTRACT FROM AUTHOR]

A discrete gyroscopic system is characterized by 2 n first-order ordinary differential equations defined by one symmetric and one skew-symmetric, which system describes the motion of a spinning body containing elastic parts. In this paper, we consider the inverse problems of such system: Given partial spectral data, find a system such that it is of the desired spectral data. The general solution of the problem is given and the best approximation solution to a pair of matrices is provided by QR-decomposition and matrix derivation. In addition, we also consider a special case in which the system operates below the lowest critical speed. The numerical examples show that the proposed method is effective. [ABSTRACT FROM AUTHOR]

Low-rank Krylov methods are one of the few options available in the literature to address the numerical solution of large-scale general linear matrix equations. These routines amount to well-known Krylov schemes that have been equipped with a couple of low-rank truncations to maintain a feasible storage demand in the overall solution procedure. However, such truncations may affect the convergence properties of the adopted Krylov method. In this paper we show how the truncation steps have to be performed in order to maintain the convergence of the Krylov routine. Several numerical experiments validate our theoretical findings. [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]

The class of { P 1 , P 2 } -Nekrasov matrices, defined in terms of permutation matrices P 1 and P 2 , is a generalization of the well-known class of Nekrasov matrices. In this paper, some computable error bounds for linear complementarity problems (LCPs) of { P 1 , P 2 } -Nekrasov matrices are given, which depend only on the entries of the involved matrices and can be used to obtain the perturbation bounds of { P 1 , P 2 } -Nekrasov matrices LCPs. Besides, some sufficient conditions ensuring that the subdirect sum of { P 1 , P 2 } -Nekrasov matrices lies in the same class are also provided. [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]

This paper is concerned with the generalized Sylvester equation A X B + C X D = E , where A, B, C, D, E are infinite size matrices with a quasi Toeplitz structure, that is, a semi-infinite Toeplitz matrix plus an infinite size compact correction matrix. Under certain conditions, an equation of this type has a unique solution possessing the same structure as the coefficient matrix does. By separating the analysis on the Toeplitz part with that on the correction part, we provide perturbation results that cater to the particular structure in the coefficient matrices. We show that the Toeplitz part is well-conditioned if the whole problem, without considering the structure, is well-conditioned. Perturbation results that are illustrated through numerical examples are applied to equations arising in the analysis of a Markov process and the 2D Poisson problem. [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]

In this paper, we study nonlinear matrix equations X p = A + ∑ i = 1 m M i T (X # B) M i and X p = A + ∑ i = 1 j M i T (X # B) M i + ∑ i = j + 1 m M i T (X - 1 # B) M i , where p, m, j are positive integers, 1 ≤ j ≤ m , A, B are n × n positive definite matrices and M i (i = 1 , 2 , 3 , ... , m) are n × n nonsingular real matrices. Based on some fixed point theorems for monotone and mixed monotone operators in ordered Banach spaces and some properties of cone, we prove that these equations always have a unique positive definite solution. In addition, an iterative sequence can be given to approximate the unique positive definite solution by employing a multi-step stationary iterative method. [ABSTRACT FROM AUTHOR]

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, we present several equivalent conditions related to the forward order law for the core inverse of two matrices, i.e., (A B) # = A # B # . In addition, we consider problems when (A B) † = A # B # , (A B) # = A # B # and (A B) # = A # B # , respectively. Thus, we study some hybrid forward order laws too. [ABSTRACT FROM AUTHOR]

The barrier of the family of centralizer codes is the length which is always n2. In our paper, we have taken codes generated by two matrices A and C of different orders n × n and k × k respectively. This family of codes are termed as intertwining codes and denoted by. Specialty of this code is the length nk which gives a new approach to characterize family of centralizer codes. In this article, we show an upper bound on the minimum distance of intertwining codes. Besides, we establish two decoding methods which can be fitted to intertwining codes as well as for any linear codes. Moreover, we have shown a condition for which a linear code can be represented as an intertwining code. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Cramer’s rule for the general solution to the system of quaternion matrix equations A 1 X B 1 = C 1 , A 2 X B 2 = C 2 , and Cramer’s rule for the general solution to the generalized Sylvester quaternion matrix equation A X B + C Y D = E , respectively. As applications, we derive the determinantal expressions for the Hermitian solutions to some quaternion matrix equations. The findings of this paper extend some known results in the literature. [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 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]

For η ∈ { i , j , k } , a real quaternion matrix A is said to be η -Hermitian if A = A η ∗ , where A η ∗ = - η A ∗ η , and A ∗ stands for the conjugate transpose of A. In this paper, we present some practical necessary and sufficient conditions for the existence of an η -Hermitian solution to a system of constrained two-sided coupled real quaternion matrix equations and provide the general η -Hermitian solution to the system when it is solvable. Moreover, we present an algorithm and a numerical example to illustrate our results. [ABSTRACT FROM AUTHOR]

The ★ -congruence Sylvester equation is the matrix equation A X + X ⋆ B = C , where A ∈ F m × n , B ∈ F n × m and C ∈ F m × m are given, whereas X ∈ F n × m is to be determined. Here, F = R or C , and ⋆ = T (transposed) or ∗ (conjugate transposed). Very recently, Satake et al. showed that under some conditions, the matrix equation for the case ⋆ = T is equivalent to the generalized Sylvester equation. In this paper, we demonstrate that the result can be extended to the case ⋆ = ∗. Through this extension, the least squares solution of the ∗ -congruence Sylvester equation may be obtained using well-researched results on the least squares solution of the generalized Sylvester equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the solution of severely ill-conditioned linear tensor equations. These kinds of equations may arise when discretizing partial differential equations in many space-dimensions by finite difference or spectral methods. The deblurring of color images is another application. We describe the tensor Golub–Kahan bidiagonalization (GKB) algorithm and apply it in conjunction with Tikhonov regularization. The conditioning of the Stein tensor equation is examined. These results suggest how the tensor GKB process can be used to solve general linear tensor equations. Computed examples illustrate the feasibility of the proposed algorithm. [ABSTRACT FROM AUTHOR]