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The semi-tensor product, which was initially proposed by Cheng et al. [An introduction to semi-tensor product of matrices and its applications. World Scientific; 2012], has been extensively applied in Boolean control networks, graph colouring, game theory, cryptographic algorithms and so on. In this article, motivated by the existing work by Yao et al. [J Franklin Inst. 2016;353:1109–1131], we further investigate the solvability of the matrix equation AXB=C with respect to semi-tensor product. The case of matrix-vector equation, in which the required unknown X be a vector, is studied first. Compatible condition for matrix dimensions, necessary and sufficient conditions and concrete solving methods are established. Based on this, the solvability of the matrix equation case, in which the unknown X be a matrix, under semi-tensor product is then studied. For each part, several elementary examples are presented to illustrate the efficiency of the results. [ABSTRACT FROM AUTHOR]

In this paper we consider the solvability of the matrix equations,with respect to the semi-tensor product. Using the concept of the semi-tensor product which was initially proposed by Cheng et al. [An introduction to Semi-Tensor product of matrices and its applications, World Scientific, 2012], firstly we investigate the matrix–vector equations,under the semi-tensor product, establish some compatible conditions for the matrices, and propose a necessary and sufficient condition for the solvability of the matrix–vector equation. We then explore the solvability of the matrix equations,under the semi-tensor product. Some elementary examples are given to show the efficiency of the results. [ABSTRACT FROM AUTHOR]

The low rank solution of the Q-weighted nearest correlation matrix problem is studied in this paper. Based on the property of Q-weighted norm and the Gramian representation, we first reformulate the Q-weighted nearest correlation matrix problem as a minimization problem of the trace function with quadratic constraints and then prove that the solution of the minimization problem the trace function is the stationary point of the original problem if it is rank-deficient. Finally, the nonmonotone spectral projected gradient method is constructed to solve them. Numerical examples illustrate that the new method is feasible and effective. Copyright © 2015 John Wiley & Sons, Ltd. [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]

Let $R\in\mathbb{R}^{m\times m}$ and $S\in\mathbb{R}^{n\times n}$ be real nontrivial symmetric involutions; i.e., R= RT= R−1≠± Im and S= ST= S−1≠± In. We say that $A\in \mathbb{R}^{m\times n}$ is ( R, S)-symmetric (( R, S)-skew symmetric) if RAS= A ( RAS=− A). Trench (Linear Algebra Appl. 389:23–31, 2004 ) has theoretically studied the minimization problems and the related approximation problems of matrix equation AZ= W for ( R, S)-symmetric matrices, using their structure properties and Moore-Penrose inverse. In this paper, we extend and develop these research in a totally different way using iterative methods. We propose two algorithms based on the idea of the classical Conjugate Gradient method (CG) and Conjugate Gradient Least Squares method (CGLS), to solve the more general equation AXB= C for ( R, S)-symmetric matrices X. Some numerical results confirm the efficiency of these algorithms. More importantly, some numerical stability analysis on the approximation problem is given combining with numerical examples, which is not given in the earlier papers. [ABSTRACT FROM AUTHOR]

The generalized Sylvester matrix equation AX + YB = C is encountered in many systems and control applications, and also has several applications relating to the problem of image restoration, and the numerical solution of implicit ordinary differential equations. In this paper, we construct a symmetric preserving iterative method, basing on the classic Conjugate Gradient Least Squares (CGLS) method, for AX + YB = C with the unknown matrices X, Y having symmetric structures. With this method, for any arbitrary initial symmetric matrix pair, a desired solution can be obtained within finitely iterate steps. The unique optimal (least norm) solution can also be obtained by choosing a special kind of initial matrix. We also consider the matrix nearness problem. Some numerical results confirm the efficiency of these algorithms. It is more important that some numerical stability analysis on the matrix nearness problem is given combined with numerical examples, which is not given in the earlier papers. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [ABSTRACT FROM AUTHOR]