The Solutions Of A Few Matrix Equations And Their Optimal Solutions

Constrained matrix equation problem is to find the solution of a matrix equation in a constrained matrix set. Different constrained conditions or different matrix equations lead to different constrained matrix equation problems. Constrained matrix equation problem is applied in many fields, such as structure design, parameter identification, main component analysis, reconnaissance, remote sensing, biology, electrics, optics, structure dynamics, molecule spectroscopy, automation theory, oscillation theory, circulation theory, linear programming, non-linear programming theory and finite element theory etc. Many different kinds of questions arisen in these fields has accelerated the development of constrained matrix equation problem and make it one of the hottest topics in numerical algebra.The constrained linear matrix equation problems in this paper mainly study the solvability conditions, optimal approximation solutions and least-square solutions of several types of linear matrix equations in several types of matrix space.1. The generalized (I,M) symmetric, generalized (I,M) anti-symmetric, generalized M symmetric and generalized M anti-symmetric solutions of matrix equation A~TXA = B are considered. The necessary and sufficient conditions for the existence and the general expressions of the solutions, the optimal approximation solutions, minimum norm solutions and some numerical experiments are obtained. Moreover, the least-squares solutions of matrix equation A~TXA = B for these matrices are also studied. the least-squares solutions, least-squares optimal approximation solutions and minimum norm least-squares solutions are obtained. The results indicate that when matrix M is invertible or has a full column rank, the generalized (I,M)(or M) symmetric or generalized (I,M)(or M) anti-symmetric solutions of matrix equation A~TXA = B are actually the symmetric or anti-symmetric solutions.2. The Hermite reflexive, anti-Hermite reflexive, Hermite anti-reflexive, anti-Hermite anti-reflexive, generalized M-Hermite, generalized M-anti-Hermite solutions of matrix equation A~HXA = B are considered. The necessary and sufficient conditions for the existence and the general expressions of the general solutions, the optimal approximation solutions and minimum norm solutions are given. Especially, the numerical examples for the generalized M-Hermite optimal approximation solution and minimum norm solution are given.3. The orthogonal-symmetric and orthogonal-anti-symmetric solutions of ma-...