Symmetric solutions of linear matrix equations by matrix decompositions

Conditions for the existence of solutions, and the general solution of linear matrix equations are given, when it is known a priori that the solution matrix has a given structure (e.g. symmetric, triangular, diagonal). This theory is subsequently extended to matrix equations that are linear in several unknown ‘structured’ matrices, and to partitioned matrix equations.

Necessary and sufficient conditions for existence and expressions for general Hermitian and nonnegative definite solutions are obtained for the following three systems of linear equations: (I) $AX = C$, (II) $AX = C,XB = D$, (III) $AXB = C$.

Bivariate matrix functions provide a unified framework for various tasks in numerical linear algebra, including the solution of linear matrix equations and the application of the Frechet derivative. In this work, we propose a novel tensorized Krylov subspace method for approximating such bivariate matrix functions and analyze its convergence. While this method is already known for some instances, our analysis appears to result in new convergence estimates and insights for all but one instance, Sylvester matrix equations.

By means of the properties of Hermitian-generalized Hamiltonian matrices, we give some necessary and sufficient conditions for the solvability of the matrix equation AX = B in Hermitian-generalized Hamiltonian matrix set $HHC^{n\times n}$. In the case where AX = B is solvable in $HHC^{n\times n}$, we derive the general representation of the solutions. We also give the expression of the solution for the corresponding optimal approximation problem.

Team-optimal closed-loop Stackelberg strategies in hierarchical control problems

The two papers in this issue have to do with matrices and sparsity—but from different points of view. Sparsity, in the first paper, means many zero elements in the matrix, while in the second paper it refers to many zero singular values, i.e., low rank. The context of the first paper, “On the Block Triangular Form of Symmetric Matrices” by Iain Duff and Bora Uçar, is the solution of linear systems of equations $Ax = b$ whose coefficient matrix A is sparse, i.e., has many zero elements. A direct method, such as Gaussian elimination, becomes more efficient if one can first permute the rows and columns of A into block triangular form. The classical method for permuting a matrix to block triangular form is due to Dulmage and Mendelsohn and dates back to 1963. The idea is to represent the matrix A as a bipartite graph whose nodes are columns and rows of A, and whose edges correspond to nonzero elements of A, and then to determine a matching of maximum cardinality in this graph. That means determining a maximal number of nonzero elements no two of which belong to the same row or column. Duff and Uçar analyze the block triangular form for a particular class of square matrices A: These matrices are structurally symmetric, i.e., their zero/nonzero structure is symmetric; and they can be structurally rank deficient, i.e., any rank deficiency is evident from the zero/nonzero structure of A. This paper illustrates nicely how graph theory can contribute to improving the efficiency of sparse matrix methods. The paper should appeal to those who need to solve large sparse linear systems, as well as those interested in graph theory. The second paper, “Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization” by Benjamin Recht, Maryam Fazel, and Pablo Parrilo, has applications to model reduction and system identification, machine learning, and image compression, just to name a few. Given a set of affine constraints for a matrix, the problem is to find a matrix of minimum rank that satisfies these constraints. The authors attack this hard nonconvex optimization problem by replacing it with a convex approximation: Instead of minimizing the rank, they minimize the sum of the singular values, the so-called nuclear norm. Nuclear norm minimization thus extends the compressed sensing framework from finding sparse vectors via $\ell_1$ minimization to finding low-rank matrices via $\ell_1$ minimization of the (vector of) singular values. The purpose of this paper is to justify mathematically why nuclear norm minimization does so well in practice. In addition to extending the restricted isometry property to matrices, the authors also discuss algorithms, computational performance, and numerical issues. This is a fascinating and well-written paper that combines results from compressed sensing, matrix theory, optimization, and probability.

This paper is a matrix iterative method presented to compute the solutions of the matrix equation, AXB=C, with unknown matrix X∈𝒮, where 𝒮 is the constrained matrices set like symmetric, symmetric-R-symmetric and (R, S)-symmetric. By this iterative method, for any initial matrix X 0∈𝒮, a solution X* can be obtained within finite iteration steps if exact arithmetics were used, and the solution X* with the minimum Frobenius norm can be obtained by choosing a special kind of initial matrix. The solution , which is nearest to a given matrix ˜X in Frobenius norm, can be obtained by first finding the minimum Frobenius norm solution of a new compatible matrix equation. The numerical examples given here show that the iterative method proposed in this paper has faster convergence and higher accuracy than the iterative methods proposed in [G.-X. Huang, F. Yin, and K. Guo, An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB=C , J. Comput. Appl. Math. 212 (2008), pp. 231–244; Y. Lei and A.-P. Liao, A minimal residual algorithm for the inconsistent matrix equation AXB=C over symmetric matrices, Appl. Math. Comput. 188 (2007), pp. 499–513; Z.-Y. Peng, An iterative method for the least squares symmetric solution of the linear matrix equation AXB=C , Appl. Math. Comput. 170 (2005), pp. 711–723; Y.-X. Peng, X.-Y. Hu, and L. Zhang, An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C , Appl. Math. Comput. 160 (2005), pp. 763–777].

SECTION 6 - Solutions of Linear Equations

For solving online the linear matrix equation AXB = C with time-varying coefficients, this paper presents a special kind of recurrent neural networks by using a design method recently proposed by Zhang et al. Compared with gradient neural networks (abbreviated as GNN, or termed as gradient-based neural networks), the resultant Zhang neural network (termed as such and abbreviated as ZNN hereafter for presentation convenience) is designed based on a matrix-valued error function, instead of a scalar-valued error function. Zhang neural network is deliberately developed in the way that its trajectory could be guaranteed to globally exponentially converge to the time-varying theoretical solution of given linear matrix equation. In addition, Zhang neural network is described by an implicit dynamics, instead of an explicit dynamics usually describing recurrent neural networks. Convergence results of Zhang neural network are presented to show the neural-network performance. In comparison, we develop and simulate the gradient neural network as well, which is exploited to solve online the time-varying linear matrix equation. Computer-simulation results substantiate the theoretical efficacy and superior performance of Zhang neural network for the online solution of time-varying linear matrix equation, especially when using a power-sigmoid activation function.

This paper deals with the problem of finding the minimum norm least-squares solution of a quite general class of coupled linear matrix equations defined over field of complex numbers. To this end, we examine a gradient-based approach and present the convergence properties of the algorithm. The highlight of the elaborated results in the current work is using a new sight of view for construction of the gradient-based algorithm which turns out that we can ignore some of the limitations assumed by the authors in the recently published works for the application of the algorithm to obtain the solution of the referred problems. To the best of our knowledge, so far, computing the optimal convergence factor of the algorithm to determine the (least-squares) solution of general complex linear matrix equations has left as a project to be investigated. In the current work, we determine the optimal convergence factor of the algorithm. Some numerical experiments are reported to illustrate the validity of the presented results.

Generating solutions of a linear equation and structure of elements of the Zelisko group

Nonlinear recurrent neural networks for finite-time solution of general time-varying linear matrix equations

Applications of the shifted Legendre polynomials expansion to the analysis and identification of the nonlinear time-delayed system, described by a memoryless nonlinear element followed by a linear plant with time delay, are studied. The system described here is assumed both controllable and observable. For analysis, by using the shifted Legendre polynomials expansion, the solution of a nonlinear state equation is reduced to the solution of a linear algebraic matrix equation. For identification, through the shifted Legendre expansions of the measured input/output data, the unknown parameters of both the linear delayed plant and the characterisation of the nonlinear element are estimated by using the least-squares method. Algorithms are presented. Numerical examples are given to illustrate the use of this approach.

Vehicles travelling at variable speed over irregular guide-ways are generally investigated by a nonstationary stochastic analysis. Up to now the variances of the vehicle response are gained from a matrix differential equation. In this paper a modification of the excitation process description is proposed which yields a substantial simplification: The resulting covariance analysis can be reduced to the solution of linear algebraic matrix equations which are solved by transformation techniques. An example demonstrates the application, the quality, and the advantages of the developed approximation method compared with the conventional exact analysis. The formulation holds for linear time-invariant multi-axle vehicle systems of arbitrary order with coloured noise excitation.

ILUBCG2: A preconditioned biconjugate gradient routine for the solution of linear asymmetric matrix equations arising from 9-point discretizations