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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]

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]

Keeping in view the latest development of anti-Hermitian matrix in mind, we construct some closed form formula for a classical system of matrix equations having anti-Hermitian nature in this paper. We give the necessary and sufficient conditions for the existence of its solution by applying the properties of matrix rank. The general solution to this system is expressed by closed formula based on generalized inverses of given matrices. The novelty of the proposed results is not only obtaining a formal representation of the solution in terms of generalized inverses but the construction of an algorithm to find its explicit expression as well. To conduct an algorithm and numerical example, it is used the determinantal representations of the Moore Penrose inverse previously obtained by one of the authors. [ABSTRACT FROM AUTHOR]

Many works on solving matrix equations are related to the complex nonlinear Yang–Baxter matrix equation A X A = X A X , where A ∈ C n × n is a given matrix and X is an unknown matrix. The Yang–Baxter matrix equation has been widely studied by its application in various fields of mathematics and physics. In this paper, we introduce an iterative method based on the Hermitian and skew-Hermitian splitting of coefficient matrix A for solving complex nonlinear Yang–Baxter matrix equation. Then, we prove the convergence of the new scheme subject to some conditions. Finally, an example is solved to discover the applicability of the new method via comparing it with some related previous methods. [ABSTRACT FROM AUTHOR]

In previous papers, the authors introduced and characterized a class of matrices called { K , s + 1 } -potent. Also, they established a method to construct these matrices. The purpose of this paper is to solve the associated inverse problem. Several algorithms are developed in order to find all involutory matrices K satisfying K A s + 1 K = A for a given matrix A ∈ C n × n and a given natural number s . The cases s = 0 and s ≥ 1 are separately studied since they produce different situations. In addition, some examples are presented showing the numerical performance of the methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new decomposition (called the core-EP decomposition in the present paper) for square matrices and give some of its applications. By applying the decomposition, we derive several characterizations of the core-EP inverse, introduce two orders – the core-EP order and the core-minus partial order, and characterize their properties. We also give some properties of the decomposition. [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, 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 introduce the weak group inverse (called as the WG inverse in the present paper) for square complex matrices of an arbitrary index, and give some of its characterizations and properties. Furthermore, we introduce two orders: one is a pre-order and the other is a partial order, and derive several characterizations of the two orders. The paper ends with a characterization of the core EP order using WG inverses.

In the present study, we are interested in solving the nonhomogeneous second-order linear difference equation with periodic coefficients of period p ≥ 2 , by bringing two new approaches enabling us to provide both analytic and combinatorial solutions to this family of equations. First, we get around the problem by converting this kind of equations to an equivalent family of nonhomogeneous linear difference equations of order p with constant coefficients. Second, we propose new expressions of the solutions of this family of equations, using our techniques of calculating the powers of product of companion matrices and some properties of generalized Fibonacci sequences. The study of the special case p = 2 is provided. And to enhance the effectiveness of our approaches, some numerical examples are discussed. [ABSTRACT FROM AUTHOR]

Abstract 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.

The T-congruence Sylvester equation is the matrix equation A X + X T B = C , where A ∈ R m × n , B ∈ R n × m and C ∈ R m × m are given, and matrix X ∈ R n × m is to be determined. The T-congruence Sylvester equation has recently attracted attention because of a relationship with palindromic eigenvalue problems. For example, necessary and sufficient conditions for the existence and uniqueness of solutions, and numerical solvers have been intensively studied. In this paper, we will show that, under a certain condition and n = m , the T-congruence Sylvester equation can be transformed into the Lyapunov equation. This may lead to further properties and efficient numerical solvers by utilizing the rich literature on the Lyapunov equation. [ABSTRACT FROM AUTHOR]

In this paper, a new iteration method is proposed for solving the complex symmetric linear systems. In theory, we show that the convergence factor or the upper bound of the spectral radius of the iteration matrix of the new method are smaller than that of the PMHSS method proposed by Bai et al. (2011). Moreover, we analyze and compare the parameter-free versions and the spectrum distributions of the preconditioned matrix of the new method and the PMHSS method. Finally, we present some numerical experiments on a few model problems to illustrate the theoretical results and show the effectiveness of our new method. [ABSTRACT FROM AUTHOR]

This paper is the second part of [15] . Taking advantage of the special structure and properties of the Hamiltonian matrix, we apply a symplectically similar transformation introduced by [18] to reduce H to a Hamiltonian Jordan canonical form J . The asymptotic analysis of the structure-preserving flows and RDEs is studied by using e J t . The convergence of the SDA as well as its rate can thus result from the study of the structure-preserving flows. A complete asymptotic dynamics of the SDA is investigated, including the linear and quadratic convergence studied in the literature [3,12,13] . [ABSTRACT FROM AUTHOR]

In this paper we solve nonlinear matrix equations of the form X δ = Q + ∑ i = 1 p ( A i ⁎ F i ( X ) A i ) r i and X δ = Q + ∑ i = 1 p ( A i ⁎ F i ( X ) A i ) r i + ∑ j = 1 q ( B j ⁎ G j ( X ) B j ) q j , where δ ∈ ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) , r i , q j ∈ [ − 1 , 1 ] , Q ∈ P ( n ) , the collection of all n × n Hermitian positive definite matrices and A i , B j 's are n × n matrices, also F i , G j 's are monotone mappings from P ( n ) into P ( n ) . Examples are given to illustrate that the equations can not be solved by previously known theorems. [ABSTRACT FROM AUTHOR]

In this paper we propose an iteration method to solve the multiple constrained least squares matrix problem. We first transform the multiple constrained least squares matrix problems into the multiple constrained matrix optimal approximation problem, and then we use the idea of Dykstra’s algorithm to derive the basic iterative pattern. We observe that we only need to solve multiple single constrained least squares matrix problems at each iteration step of the proposed algorithm. We give a numerical example to illustrate the effectiveness of the proposed method to solve the original problems. Also, we give an example to illustrate that the method proposed by Escalante and Li to solve the single constrained least squares matrix problem is not correct. [ABSTRACT FROM AUTHOR]

An explicit solution to the partial eigenvalue assignment problem of high order control system is presented by the method of receptance. Conventional methods, e.g. finite elements, are known to contain inaccuracies and assumptions that may hinder the calculations. An alternative approach was given by Ram and Mottershead [1] in the form of receptances, typically available from a modal test. This paper generalizes the earlier work on partial assignment that is applicable to multi-input delayed system without use of the Sherman–Morrison formula. The results of our numerical experiments support the validity of our proposed method. [ABSTRACT FROM AUTHOR]

In this paper we introduce an isospectral matrix flow (Lax flow) that preserves some structures of an initial matrix. This flow is given by d A d t = [ A u − A l , A ] , A ( 0 ) = A 0 , where A is a real n × n matrix (not necessarily symmetric), [ A , B ] = A B − B A is the matrix commutator (also known as the Lie bracket), A u is the strictly upper triangular part of A and A l is the strictly lower triangular part of A . We prove that if the initial matrix A 0 is staircase, so is A ( t ) . Moreover, we prove that this flow preserves the certain positivity properties of A 0 . Also we prove that if the initial matrix A 0 is totally positive or totally nonnegative with non-zero codiagonal elements and distinct eigenvalues, then the solution A ( t ) converges to a diagonal matrix while preserving the spectrum of A 0 . Some simulations are provided to confirm the convergence properties. [ABSTRACT FROM AUTHOR]

This paper presents new results on the prediction of linear processes in function spaces. The autoregressive Hilbertian process framework of order one (ARH(1) framework) is adopted. A component-wise estimator of the autocorrelation operator is derived from the moment-based estimation of its diagonal coefficients with respect to the orthogonal eigenvectors of the autocovariance operator, which are assumed to be known. Mean-square convergence to the theoretical autocorrelation operator is proved in the space of Hilbert–Schmidt operators. Consistency then follows in that space. Mean absolute convergence, in the underlying Hilbert space, of the ARH(1) plug-in predictor to the conditional expectation is obtained as well. A simulation study is undertaken to illustrate the large-sample behavior of the formulated component-wise estimator and predictor. Additionally, alternative component-wise (with known and unknown eigenvectors), regularized, wavelet-based penalized, and nonparametric kernel estimators of the autocorrelation operator are compared with the one presented here, in terms of prediction. [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]

Between 1970 and 1982 Hans Schneider and co-authors produced a number of results regarding matrix scalings. They demonstrated that a matrix has a diagonal similarity scaling to any matrix with entries in the subgroup generated by the cycle weights of the associated digraph, and that a matrix has an equivalent scaling to any matrix with entries related to the weights of cycles in an associated bipartite graph. Further, given matrices A and B , they produced a description of all diagonal X such that X − 1 A X = B . In 2005 Butkovič and Schneider used max-algebra to give a simple and efficient description of this set of scalings. In this paper we focus on the additive group of integers, and work in the max-plus algebra to give a full description of all scalings of a real matrix A to any integer matrix. We do this for four types of scalings; beginning with the familiar X − 1 A X , XAY and XAX scalings and finishing with a new scaling which we call a signed similarity scaling. This is a scaling of the form XAY where we specify for each row i , either x i = y i or x i = − y i . In all of our results we use necessary and sufficient conditions for existence which are based on integer weighted cycles in the associated digraph, or associated bipartite graph, of the matrix. [ABSTRACT FROM AUTHOR]

Tensor computation is intensive and difficult. Invariably, a vital component is the truncation of tensors, so as to control the memory and associated computational requirements. Various tensor toolboxes have been designed for such a purpose, in addition to transforming tensors between different formats. In this paper, we propose a simple Q-less QR truncation technique for tensors { x ( i ) } with x ( i ) ∈ R n 1 × ⋯ × n d in the simple and natural Kronecker product form. It generalizes the QR decomposition with column pivoting, adapting the well-known Gram–Schmidt orthogonalization process. The main difficulty lies in the fact that linear combinations of tensors cannot be computed or stored explicitly. All computations have to be performed on the coefficients α i in an arbitrary tensor v = ∑ i α i x ( i ) . The orthonormal Q factor in the QR decomposition X ≡ [ x ( 1 ) , ⋯ , x ( p ) ] = Q R cannot be computed but expressed as X R − 1 when required. The resulting algorithm has an O ( p 2 d n ) computational complexity, with n = max ⁡ n i . Some illustrative examples in the numerical solution of tensor linear equations are presented. [ABSTRACT FROM AUTHOR]

Given a Laurent polynomial with matrix coefficients that is positive semi-definite over the unit circle in the complex plane, the Fejér–Riesz theorem asserts that it can always be factorized as the product of a polynomial with matrix coefficients and its adjoint. This paper exploits such a factorization in its simplest form of degree one and its relationship with the nonlinear matrix equation X + A ⁎ X − 1 A = Q . In particular, the nonlinear equation can be recast as a linear Sylvester equation subject to unitary constraint. The Sylvester equation is readily obtainable from hermitian eigenvalue computation. The unitary constraint can be enforced by a hybrid of a straightforward alternating projection for low precision estimation and a coordinate-free Newton iteration for high precision calculation. This approach offers a complete parametrization of all solutions and, in contrast to most existent algorithms, makes it possible to find all solutions if so desired. [ABSTRACT FROM AUTHOR]

This article discusses an approach to solving large-scale algebraic Riccati equations (AREs) by computing a low-dimensional stable invariant subspace of the associated Hamiltonian matrix. We give conditions on AREs to admit solutions of low numerical rank and show that these can be approximated via Hamiltonian eigenspaces. We discuss strategies on choosing the proper eigenspace that yields a good approximation, and different formulas for building the approximation itself. Similarities of our approach with several other methods for solving AREs are shown: closely related are the projection-type methods that use various Krylov subspaces and the qADI algorithm. The aim of this paper is merely to analyze the possibilities of computing approximate Riccati solutions from low-dimensional subspaces related to the corresponding Hamiltonian matrix and to explain commonalities among existing methods rather than providing a new algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we study partial orders in terms of the core-nilpotent decomposition. We derive some characterizations of the C-N partial ordering, create a new partial ordering (the G-Drazin partial ordering) which is a generalization of C-N partial ordering, and give some properties and characterizations of the G-Drazin partial ordering. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]