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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]

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 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 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 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]

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 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]

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]

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]

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]

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]

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, 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]

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, 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]

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]

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]

In this paper,we introduce the weak group matrix defined by the one commutable with its weak group inverse, and consider properties and characterizations of the matrix by applying the core-EP decomposition. In particular,the set of weak group matrices is more inclusive than that of group matrices. We also derive some characterizations of p-EP matrices and i-EP matrices. [ABSTRACT FROM AUTHOR]

In this paper, we consider two kinds of nonlinear matrix equations X + ∑ i = 1 m B i ∗ X t i B i = I (0 < t i < 1) and X s - ∑ i = 1 m A i ∗ X p i A i = I (p i > 1 , s ≥ 1) . By means of the integral representation of matrix functions, properties of Kronecker product and the monotonic p-concave operator fixed point theorem, we derive necessary conditions and sufficient conditions for the existence and uniqueness of the Hermitian positive definite solution for the matrix equations. We also obtain some properties of the Hermitian positive definite solutions, the bounds of the determinant's sum for A i ∗ A i and the spectral radius of A i . [ABSTRACT FROM AUTHOR]

In many areas, there arise linear systems of the form A B ⊤ B - D x y = f g , where A ∈ R n × n , D ∈ R p × p are symmetric and positive semi-definite and B ∈ R p × n. In this paper, some simple criteria for this special linear systems to have solutions and the unique solution are provided, and the solvability conditions are expressed by A , B , D , f and g . [ABSTRACT FROM AUTHOR]

Given a closed convex cone C in a finite dimensional real Hilbert space H, a weakly homogeneous map f : C → H is a sum of two continuous maps h and g, where h is positively homogeneous of degree γ ( ≥ 0 ) on C and g (x) = o (| | x | | γ) as | | x | | → ∞ in C. Given such a map f, a nonempty closed convex subset K of C, and a q ∈ H , we consider the variational inequality problem, VI (f , K , q) , of finding an x ∗ ∈ K such that ⟨ f (x ∗) + q , x - x ∗ ⟩ ≥ 0 for all x ∈ K. In this paper, we establish some results connecting the variational inequality problem VI (f , K , q) and the cone complementarity problem CP (f ∞ , K ∞ , 0) , where f ∞ : = h is the homogeneous part of f and K ∞ is the recession cone of K. We show, for example, that VI (f , K , q) has a nonempty compact solution set for every q when zero is the only solution of CP (f ∞ , K ∞ , 0) and the (topological) index of the map x ↦ x - Π K ∞ (x - G (x)) at the origin is nonzero, where G is a continuous extension of f ∞ to H. As a consequence, we generalize a complementarity result of Karamardian (J Optim Theory Appl 19:227–232, 1976) formulated for homogeneous maps on proper cones to variational inequalities. The results above extend some similar results proved for affine variational inequalities and for polynomial complementarity problems over the nonnegative orthant in R n . As an application, we discuss the solvability of nonlinear equations corresponding to weakly homogeneous maps over closed convex cones. In particular, we extend a result of Hillar and Johnson (Proc Am Math Soc 132:945–953, 2004) on the solvability of symmetric word equations to Euclidean Jordan algebras. [ABSTRACT FROM AUTHOR]

This paper investigates the randomized version of the Kaczmarz method to solve linear systems in the case where the adjoint of the system matrix is not exact—a situation we refer to as "mismatched adjoint". We show that the method may still converge both in the over- and underdetermined consistent case under appropriate conditions, and we calculate the expected asymptotic rate of linear convergence. Moreover, we analyze the inconsistent case and obtain results for the method with mismatched adjoint as for the standard method. Finally, we derive a method to compute optimized probabilities for the choice of the rows and illustrate our findings with numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, some errors are pointed out and the correct results are given for the paper Axelsson et al. (J Comput Appl Math 280:141-157, 2015). Then, a class of general augmentation block preconditioners for solving generalized saddle point systems with singular (1, 1) blocks are considered. Results concerning the eigenvalue distribution and forms of the eigenvectors of the preconditioned generalized saddle point matrix are presented. These results extend previous one in the literature. [ABSTRACT FROM AUTHOR]

In this paper, the parametric matrix equation A( p) X = B( p) whose elements are linear functions of uncertain parameters varying within intervals are considered. In this matrix equation A( p) and B( p) are known m-by- m and m-by- n matrices respectively, and X is the m-by- n unknown matrix. We discuss the so-called AE-solution sets for such systems and give some analytical characterizations for the AE-solution sets and a sufficient condition under which these solution sets are bounded. We then propose a modification of Krawczyk operator for parametric systems which causes reduction of the computational complexity of obtaining an outer estimation for the parametric united solution set, considerably. Then we give a generalization of the Bauer-Skeel and the Hansen-Bliek-Rohn bounds for enclosing the parametric united solution set which also enables us to reduce the computational complexity, significantly. Also some numerical approaches based on Gaussian elimination and Gauss-Seidel methods to find outer estimations for the parametric united solution set are given. Finally, some numerical experiments are given to illustrate the performance of the proposed methods. [ABSTRACT FROM AUTHOR]

A complex matrix X is called an $\{i,\ldots, j\}$-inverse of the complex matrix A, denoted by $A^{(i,\ldots, j)}$, if it satisfies the ith, ..., jth equations of the four matrix equations (i) $AXA = A$, (ii) $XAX=X$, (iii) $(AX)^{*} = AX$, (iv) $(XA)^{*} = XA$. The eight frequently used generalized inverses of A are $A^{\dagger}$, $A^{(1,3,4)}$, $A^{(1,2,4)}$, $A^{(1,2,3)}$, $A^{(1,4)}$, $A^{(1,3)}$, $A^{(1,2)}$, and $A^{(1)}$. The $\{i,\ldots, j\}$-inverse of a matrix is not necessarily unique and their general expressions can be written as certain linear or quadratic matrix-valued functions that involve one or more variable matrices. Let A and B be two complex matrices such that the product AB is defined, and let $A^{(i,\ldots ,j)}$ and $B^{(i,\ldots,j)}$ be the $\{i,\ldots, j\}$-inverses of A and B, respectively. A prominent problem in the theory of generalized inverses is concerned with the reverse-order law $(AB)^{(i,\ldots,j)} = B^{(i,\ldots,j)}A^{(i,\ldots,j)}$. Because the reverse-order products $B^{(i,\ldots,j)}A^{(i,\ldots,j)}$ are usually not unique and can be written as linear or nonlinear matrix-valued functions with one or more variable matrices, the reverse-order laws are in fact linear or nonlinear matrix equations with multiple variable matrices. Thus, it is a tremendous and challenging work to establish necessary and sufficient conditions for all these reverse-order laws to hold. In order to make sufficient preparations in characterizing the reverse-order laws, we study in this paper the algebraic performances of the products $B^{(i,\ldots,j)}A^{(i,\ldots,j)}$. We first establish 126 analytical formulas for calculating the global maximum and minimum ranks of $B^{(i,\ldots,j)}A^{(i,\ldots,j)}$ for the eight frequently used $\{i,\ldots, j\}$-inverses of matrices $A^{(i,\ldots,j)}$ and $B^{(i,\ldots,j)}$, and then use the rank formulas to characterize a variety of algebraic properties of these matrix products. [ABSTRACT FROM AUTHOR]

In this paper we consider the nearness symmetric solution of the matrix equation AXB = C to a given matrix $$\tilde{X}$$ in the sense of the Frobenius norm. By discussing equivalent form of the considered problem, we derive some necessary and sufficient conditions for the matrix $$X^{*}$$ is a solution of the considered problem. Based on the idea of the alternating variable minimization with multiplier method, we propose two iterative methods to compute the solution of the considered problem, and analyze the global convergence results of the proposed algorithms. Numerical results illustrate the proposed methods are more effective than the existing two methods proposed in Peng et al. (Appl Math Comput 160:763-777, ) and Peng (Int J Comput Math 87: 1820-1830, ). [ABSTRACT FROM AUTHOR]

The least squares problems have wide applications in inverse Sturm-Liouville problem, particle physics and geology, inverse problems of vibration theory, control theory, digital image and signal processing. In this paper, we discuss the solution of the operator least squares problem. By extending the conjugate gradient least squares method, we propose an efficient matrix algorithm for solving the operator least squares problem. The matrix algorithm can find the solution of the problem within a finite number of iterations in the absence of round-off errors. Some numerical examples are given to illustrate the effectiveness of the matrix algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we establish some conditions for the existence and the representations for the common ( P, Q) generalized reflexive and anti-reflexive solutions of matrix equations $$AX=B$$ and $$XC=D$$ , where P and Q are two generalized reflection matrices. Moreover, in corresponding solution set of the equations, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm has been presented. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new conditions for the existence and the representations for the $$(P, Q)$$ generalized reflexive and anti-reflexive solutions to matrix equation $$AX = B$$ with respect to the generalized reflection matrix dual $$(P, Q)$$ . Moreover, in corresponding solution sets of the equation, the explicit expressions of the nearest matrix to a given matrix in the Frobenius norm have been provided. [ABSTRACT FROM AUTHOR]

In this paper, by extending the classical Newton method, we present the generalized Newton method (GNM) with high-order convergence for solving a class of large-scale linear complementarity problems, which is based on an additional parameter and a modulus-based nonlinear function. Theoretically, the performance of high-order convergence is analyzed in detail. Some numerical experiments further demonstrate the efficiency of the proposed new method. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the following rank-constrained matrix approximation problem in the Frobenius norm: ‖ C - A X ‖ = min subject to rk C 1 - A 1 X = b , where b is an appropriate chosen nonnegative integer. We solve the problem by applying the classical rank-constrained matrix approximation, the singular value decomposition, the quotient singular value decomposition and generalized inverses, and get two general forms of the least squares solutions. [ABSTRACT FROM AUTHOR]