Research On The Solutions Of Two Classes Of Matrix Equations Over Orthogonal Matrices

The problem of solving linear matrix equations has been a hot topic in the field of numerical algebra in recent years. It is applied in many fields, such as parameter identification, automation theory, reconnaissance, remote sensing, etc. Due to differ-ent fields, different background, different constrained conditions or different matrix equations, many different problems of solving matrix equations and the corresponding optimal approximation problems can be proposed.In this paper, the solutions of the linear matrix equations AX=B on the set of orthogonal matrices are studied systematically. The necessary and sufficient conditions for the solvability of the linear matrix equation AX=B, with centro-symmetric or bisymmetric orthogonal matrix constraints are given, and the general expressions of the solutions as well as the optimal approximation solutions are obtained. Furthermore, in order to verify the correctness of the conclusions, some algorithms and numerical experiments are reported. Finally, the necessary and sufficient conditions for solvability of the linear matrix equation AXB=D, with the orthogonal solutions are obtained. The main works and results are as follows:1. The centro-symmetric orthogonal solutions as well as the optimal approxi-mation solutions of the linear matrix equation AX=B are studied. By studying the structure and properties of centro-symmetric orthogonal matrix, the necessary and suf-ficient conditions for the solvability and the general expressions of the solutions as well as the optimal approximation solutions are obtained by means of the singular value decomposition and spectral decomposition. Finally, in order to verify the correctness of the conclusions, some algorithms and numerical experiments are reported.2. The bisymmetric orthogonal solutions as well as the optimal approximation solutions of the linear matrix equation AX=B are studied. By studying the struc-ture and properties of bisymmetric orthogonal matrix, the necessary and sufficient conditions for the solvability and the general expressions of the solutions as well as the optimal approximation solutions are obtained, by means of the singular value decom-position and spectral decomposition. Finally, in order to verify the correctness of the conclusions, some algorithms and numerical experiments are reported.3. The orthogonal solutions of the linear matrix equation AXB=D are studied. The problem of solving the equation AXB=D can be transformed into another equation equivalently, and by investigating the solvability of this new equation, the necessary and sufficient conditions for the solvability of the equation AXB=D are obtained.