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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 investigate the nonlinear matrix equations X + A * X q A = Q ( 0 < q ≤ 1 ) and X − A * X q A = Q ( q > 1 ) . Necessary and sufficient conditions for the (unique) existence of Hermitian positive definite solutions of these equations are derived. Effective iterative algorithms are also provided to obtain the unique solution of X + A * X q A = Q ( 0 < q ≤ 1 ) and the minimal and maximal Hermitian positive definite solutions of X − A * X q A = Q ( q > 1 ) . Numerical examples are included to illustrate the efficiency of the presented iteration algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we first establish some necessary and sufficient conditions for the existence of Hermitian solution of A X A * = B subject to CXC * ≥ D , where B and D are Hermitian matrices. Furthermore, a general expression for this Hermitian constrained solution is derived, several special cases are also considered. [ABSTRACT FROM AUTHOR]

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 discuss the following two minimum rank matrix approximation problems in the spectral norm: (i) For given A = A H ∈ C m × m , B ∈ C m × n , determining X ∈ S 1 , such that rank ( X ) = min Y ∈ S 1 rank ( Y ) , S 1 = { Y = Y H ∈ C n × n : ‖ A − B Y B H ‖ 2 = min } . (ii) For given A = − A H ∈ C m × m , B ∈ C m × n , determining X ∈ S 2 , such that rank ( X ) = min Y ∈ S 2 rank ( Y ) , S 2 = { Y = − Y H ∈ C n × n : ‖ A − B Y B H ‖ 2 = min } . By applying the norm-preserving dilation theorem, the Hermitian-type (skew-Hermitian-type) generalized singular value decomposition (HGSVD, SHGSVD), we characterize the expressions of the minimum rank and derive a general form of minimum rank (skew) Hermitian solutions to the matrix approximation problem. [ABSTRACT FROM AUTHOR]

For the large-scale nonlinear matrix equations with low rank structure, the well-developed doubling algorithm in low rank form (DA-LR) is known as an efficient method to compute the stabilizing solution. By further analyzing the global efficiency index constructed in this paper, we propose a tripling algorithm in low rank form (TA-LR) from two points of view, the cyclic reduction and the symplectic structure preservation. The new presented algorithm shares the same pre-processing complexity with that of DA-LR, but can attain the prescribed normalized residual level within less iterations by only consuming some negligible iteration time as an offset. Under the solvability condition, the proposed algorithm is demonstrated to inherit a cubic convergence and is capable of delivering errors from the current iteration to the next with the same order. Numerical experiments including some from nano research show that the TA-LR is highly efficient to compute the stabilizing solution of large-scale nonlinear matrix equations with low rank structure. [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]

This paper deals with a characterization of the solution set of algebraic Riccati equation (ARE) (over reals) for both controllable and uncontrollable systems. We characterize all solutions using simple linear algebraic arguments. It turns out that solutions of ARE of maximal rank have lower rank solutions encoded within it. We demonstrate how these lower rank solutions are encoded within the full rank solution and how one can retrieve the lower rank solutions from the maximal rank solution. We characterize situations where there are no full rank solutions to the ARE. We also characterize situations when the number of solutions to the ARE is finite, when they are infinite and when they are bounded. We also explore the poset structure on the solution set of ARE, which in some specific cases turns out to be a lattice which is isomorphic to lattice of invariant subspaces of a certain matrix. We provide several examples that bring out the essence of these results. [ABSTRACT FROM AUTHOR]

In this paper, we extend the generalized product-type bi-conjugate gradient (GPBiCG) method for solving the generalized Sylvester-conjugate matrix equations A 1 X B 1 + C 1 Y ¯ D 1 = S 1 , A 2 X ¯ B 2 + C 2 Y D 2 = S 2 by the real representation of the complex matrix and the properties of Kronecker product and vectorization operator. Some numerical experiments demonstrate that the introduced iteration approach is efficient. [ABSTRACT FROM AUTHOR]

The generalized Sylvester matrix equation ∑ i = 1 s A i X B i + ∑ j = 1 t C j Y D j = E with unknown matrices X and Y is encountered in many system and control applications. In this paper, a direct method is established to solve the least-squares symmetric and skew-symmetric solutions of the equation by using the Kronecker product and the generalized inverses and, the expression of the solution set S are provided. Moreover, an optimal approximation between a given matrix pair and the affine subspace S is discussed, and an explicit formula for the unique optimal approximation solution is presented. Finally, two numerical examples are given which demonstrate that the introduced algorithm is quite efficient. [ABSTRACT FROM AUTHOR]

Let H m × n be the set of all m × n matrices over the real quaternion algebra H = { c 0 + c 1 i + c 2 j + c 3 k ∣ i 2 = j 2 = k 2 = i j k = − 1 , c 0 , c 1 , c 2 , c 3 ∈ R } . A ∈ H n × n is known to be η -Hermitian if A = A η * = − η A * η , η ∈ { i , j , k } and A * means the conjugate transpose of A . We mention some necessary and sufficient conditions for the existence of the solution to the system of real quaternion matrix equations including η -Hermicity A 1 X = C 1 , A 2 Y = C 2 , Y B 2 = D 2 , Y = Y η * , A 3 Z = C 3 , Z B 3 = D 3 , Z = Z η * , A 4 X + ( A 4 X ) η * + B 4 Y B 4 η * + C 4 Z C 4 η * = D 4 , and also construct the general solution to the system when it is consistent. The outcome of this paper diversifies some particular results in the literature. Furthermore, we constitute an algorithm and a numerical example to comprehend the approach established in this treatise. [ABSTRACT FROM AUTHOR]

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]

The aim of this paper is to introduce the construction of the Bézout matrix of two univariate polynomials given by values in the Hermite interpolation basis, namely the confluent Bézout matrix. Moreover, if such polynomials have exactly one common simple zero, we describe how to compute it from the null space of the confluent Bézout matrix. [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]

In this paper, we consider the problem of factorizing the n × n matrix J n of all ones into the n × n binary matrices. We show that under some conditions on the factors, these are isomorphic to a row permutation of a De Bruijn matrix. Moreover, we consider in particular the binary roots of J n , i.e. the binary solutions to A m = J n . On the one hand, we prove that any binary root with minimum rank is isomorphic to a row permutation of a De Bruijn matrix whose row permutation is represented by a block diagonal matrix. On the other hand, we partially solve Hoffman's open problem of characterizing the binary solutions to A 2 = J n by providing a characterization of the binary solutions to A 2 = J n with minimum rank. Finally, we provide a class of roots which are isomorphic to a De Bruijn matrix. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to solve the problem of finding a geodesic that best fits a given set of time-labelled points on the Grassmann manifold. To achieve this goal, we first derive a very useful simplified formula for the geodesic arc joining two points on the Grassmannian depending explicitly only on the given points. This allows to simplify the expression for the geodesic distance, which is crucial to generalize the fitting problem, and is also used to obtain a simpler characterization of the geometric mean of a finite set of points lying on the Grassmannian, where the given points enter explicitly. [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 derive some equivalent conditions for block two-by-two matrices to be nonsingular in an elementary way. [ABSTRACT FROM AUTHOR]

The semigroup of Hamiltonians acting on the cone of positive definite matrices via linear fractional transformations satisfies the Birkhoff contraction formula for the Thompson metric. In this paper we describe the action of the Hamiltonians lying in the boundary of the semigroup. This involves in particular a construction of linear transformations leaving invariant the cone of positive definite matrices (strictly positive linear mappings) parameterized over all square matrices. Its invertibility and relation to the Lyapunov and Stein operators are investigated in detail. In particular, it is shown that each of these linear transformations commutes with the corresponding Lyapunov operator and contracts the Thompson metric. [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]

One of the aims of this paper is to provide a short survey on the-graded, the symmetric and the left (right) generalizations of the classical determinant theory for square matrices with entries in an arbitrary (possibly non-commutative) ring. This will put us in a position to give a motivation for our main results. We use the preadjoint matrix to exhibit a general trace expression for the symmetric determinant. The symmetric version of the classical Newton trace formula is also presented in thecase. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for the existence of a solution to the system of equationsandfor adjointable operators between Hilbert-modules, and obtain an expression of the general solution to this system. As an application, we consider the G-Hermitian solution to the system of equationsandfor adjointable operators between Hilbert-modules. Some results of this paper are new even in the finite-dimensional case. [ABSTRACT FROM PUBLISHER]

In this paper, we consider a new nonlinear matrix equation, which is a special stochastic algebraic Riccati equation arising in stochastic control theory. Based on Brower’s fixed point theorem and Bhaskar–Lakshmikantham’s fixed point theroem, we derive some new sufficient conditions for the existence of a (unique) positive definite solution, which are easy to check. An iterative method is proposed to compute the unique positive definite solution. The error estimation of the iterative method is also given. Numerical examples show that the iterative method is feasible. [ABSTRACT FROM PUBLISHER]

In this paper, we obtain the optimal multiplicative perturbation bounds for the Moore–Penrose inverse under the Frobenius norm, the unitarily invariant norm and the-norm, respectively. These bounds always improve the previous bounds. Furthermore, all these bounds can be applied to the weighted Moore–Penrose inverse. [ABSTRACT FROM PUBLISHER]

In this paper, the Hermitian positive-definite solutions of the matrix equationXs+A*X−tA=Qare considered. New necessary and sufficient conditions for the equation to have a Hermitian positive-definite solution are derived. In particular, whenAis singular, a new estimate of Hermitian positive-definite solutions is obtained. In the end, based on the fixed point theorem, an iterative algorithm for obtaining the positive-definite solutions of the equation withQ=Iis discussed. The error estimations are found. [ABSTRACT FROM AUTHOR]

Abstract: The main aim of this paper is to provide a higher-order convergent iterative method in order to calculate the generalized inverse of a given matrix. We extend the iterative method proposed in Li et al. [W.G. Li, Z. Li, A family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non-square matrix, Applied Mathematics and Computation 215 (2010) 3433-3442] to compute the -inverse, generalized inverse -inverse and -inverse. Moreover, we modify the iterative method to compute the generalized inverse of Toeplitz matrices by using the displacement theory. [Copyright &y& Elsevier]

The purpose of this paper is to show how to use the Gauss map to produce efficient triangulations of parametric surfaces. The central idea is to invert canonical triangulations of the sphere using the Gauss map. The main problem is that the inverse of the Gauss map is not in general well defined. We show how to overcome this problem and formulate an algorithm to generate the desired triangulation. [ABSTRACT FROM PUBLISHER]

In this paper, we present an iterative algorithm for solving the following coupled Sylvester-transpose matrix equationsover the generalized centro-symmetric matrix group (X1,X2, …,Xq). The solvability of the problem can be determined by the proposed algorithm, automatically. If the coupled Sylvester-transpose matrix equations are consistent over the generalized centro-symmetric matrices, then a generalized centro-symmetric solution group can be obtained within finite iterative steps for any initial generalized centro-symmetric matrix group in the exact arithmetic. Furthermore, it is shown that the least-norm generalized centro-symmetric solution group of the coupled Sylvester-transpose matrix equations can be computed by choosing an appropriate initial iterative matrix group. Moreover, the optimal approximate generalized centro-symmetric solution group to a given arbitrary matrix group (V1,V2, …,Vq) can be derived by finding the least-norm generalized centro-symmetric solution group of a new coupled Sylvester-transpose matrix equations. Finally, some numerical results are given to illustrate the validity and practicability of the theoretical results established in this work. [ABSTRACT FROM PUBLISHER]

In this paper, we present an iterative method to compute the symmetric solutions of the matrix inequalityAX≥Bin least-squares sense. The presented iteration method can be also used to compute the symmetric solutions of the matrix equationAX=Band the consistent matrix inequalityAX≥B. Numerical experiments are given to illustrate the usefulness of the proposed approach. [ABSTRACT FROM PUBLISHER]

In the present paper, we investigate the quaternion matrix equation X−AXF=C and X−A[Xtilde] F=C. For convenience, we named the quaternion matrix equations X−AXF=C and X−A[Xtilde] F=C as quaternion Stein matrix equation and quaternion Stein-conjugate matrix equation. Based on the Kronecker map and complex representation of a quaternion matrix, we give the solution expressions of the quaternion Stein matrix equation and quaternion Stein-conjugate matrix equation. Through these expressions, we can easily obtain the solution of the above two equations. In order to compare the direct algorithm with the indirect algorithm, we propose an example to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we present the explicit expressions of the perturbation of the Drazin inverse under different conditions. Also, we give the upper bounds of ‖(A+E)D−A D‖ P /‖A D‖ P for these cases. [ABSTRACT FROM PUBLISHER]

In this paper, combining the equivalent form of the unified coupled algebraic Riccati equation (UCARE) with the eigenvalue inequalities of a matrix's sum and product, using the properties of an M-matrix and its inverse matrix, we offer new lower and upper matrix bounds for the solution of the UCARE. Furthermore, applying the derived lower and upper matrix bounds and a fixed-point theorem, an existence uniqueness condition of the solution of the UCARE is proposed. Then, we propose a new fixed-point iterative algorithm for the solution of the UCARE. Finally, we present a corresponding numerical example to demonstrate the effectiveness of our results. [ABSTRACT FROM PUBLISHER]

Abstract: This paper studies the solvability, existence of unique solution, closed-form solution and numerical solution of matrix equation with and where is the unknown. It is proven that the solvability of these equations is equivalent to the solvability of some auxiliary standard Stein equations in the form of where the dimensions of the coefficient matrices and are the same as those of the original equation. Closed-form solutions of equation can then be obtained by utilizing standard results on the standard Stein equation. On the other hand, some generalized Stein iterations and accelerated Stein iterations are proposed to obtain numerical solutions of equation . Necessary and sufficient conditions are established to guarantee the convergence of the iterations. [Copyright &y& Elsevier]

In this paper, an iterative method is presented to solve the following constrained minimum Frobenius norm residual problem: [image omitted] where [image omitted] is a linear operator from Rm×n onto [image omitted] , [image omitted] , [image omitted] is a linear self-conjugate involution operator. By this method, for any initial matrix [image omitted] , a solution can be obtained in finite iteration steps in the absence of roundoff errors. The least norm solution can be derived when an appropriate initial matrix is chosen. In addition, the optimal approximation solution in the solution set of the above problem to a given matrix can also be derived by this method. Several numerical examples are given to show the efficiency of the proposed iterative method. [ABSTRACT FROM AUTHOR]

This paper is devoted to the perturbation analysis for the generalized Sylvester equation and the generalized Lyapunov equation. The explicit expressions and upper bounds of normwise, mixed and componentwise condition numbers for these equations are presented. The results are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we obtain the additive and multiplicative perturbation bounds for the Moore-Penrose inverse under the unitarily invariant norm and the Q - norm, which improve the corresponding ones in [P.Å. Wedin, Perturbation theory for pseudo-inverses, BIT 13(1973)217–232]. [ABSTRACT FROM AUTHOR]

In this paper, an iterative method is presented for finding the bisymmetric solutions of a pair of consistent matrix equations A1XB1=C1, A2XB2=C2, by which a bisymmetric solution can be obtained in finite iteration steps in the absence of round-off errors. Moreover, the solution with least Frobenius norm can be obtained by choosing a special kind of initial matrix. In the solution set of the matrix equations, the optimal approximation bisymmetric solution to a given matrix can also be derived by this iterative method. The efficiency of the proposed algorithm is shown by some numerical examples. [ABSTRACT FROM AUTHOR]

This paper is concerned with minimal norm least squares solution to general linear matrix equations including the well-known Lyapunov matrix equation and Sylvester matrix equation as special cases. Two iterative algorithms are proposed to solve this problem. The first method is based on the gradient search principle for solving optimization problem and the second one can be regarded as its dual form. For both algorithms, necessary and sufficient conditions guaranteeing the convergence of the algorithms are presented. The optimal step sizes such that the convergence rates of the algorithms are maximized are established in terms of the singular values of some coefficient matrix. It is believed that the proposed methods can perform important functions in many analysis and design problems in systems theory. [ABSTRACT FROM AUTHOR]

Abstract: The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation , as well as the extremal inertias of a partial block Hermitian matrix. [Copyright &y& Elsevier]

This paper is concerned with numerical solutions to general linear matrix equations including the well-known Lyapunov matrix equation and Sylvester matrix equation as special cases. Gradient based iterative algorithm is proposed to approximate the exact solution. A necessary and sufficient condition guaranteeing the convergence of the algorithm is presented. A sufficient condition that is easy to compute is also given. The optimal convergence factor such that the convergence rate of the algorithm is maximized is established. The proposed approach not only gives a complete understanding on gradient based iterative algorithm for solving linear matrix equations, but can also be served as a bridge between linear system theory and numerical computing. Numerical example shows the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the nonlinear matrix equation is investigated. Based on the fixed-point theory, the existence and the uniqueness of the positive definite solution are studied. An effective iterative method to obtain the unique positive definite solution is established given . In addition, some computable estimates of the unique positive definite solution are derived. Finally, numerical examples are given to illustrate the effectiveness of the algorithm and the perturbation estimates. [Copyright &y& Elsevier]

Given two polynomials, this paper is devoted to describing the natural relation between the Euclidean algorithm and the block LU factorization of the Hankel and Bezout matrices associated to such polynomials. [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]