Solutions Of Some Constrained Systems Of Matrix Equations And Their Least Squares Problems

The constrained linear matrix equations or systems of matrix equations and the re-lated least squares problems have been of interest for many applications, including con-trol theory, vibration theory, geology, particle physics, signal processing, finite elements and so on. In this dissertation, we respectively investigate some constrained solutions and the least squares solutions of some matrix equations or systems of matrix equations over the real field or complex field, and obtain many important research results. The results obtained enrich and develop the matrix algebra theory.The dissertation is divided into6chapters.In Chapter1, the research background and the progresses of the matrix equations as well as the work of the dissertation we have done are presented. In addition, some symbols are presented.In Chapter2, the decompositions of the generalized bi (skew-) symmetric matrices are presented for the first time. Then, using the decompositions, the solvability condi-tions for the existence and the explicit expressions of the generalized bi (skew-) symmet-ric solutions, the extremal ranks and the generalized bi (skew-) symmetric least squares solutions of the matrix equation AX=B are obtained, respectively.In Chapter3, by using the theorem of the special matrix decomposition, we respec-tively discuss the necessary and sufficient conditions for the existence, the expression for the (anti-)Hermitian generalized (anti-)Hamiltonian solutions, the optimal approximate solution and the (anti-)Hermitian generalized (anti-)Hamiltonian least squares solutions of the system of complex matrix equations AX=B, XC=D for the first time.In Chapter4, by applying the theorem of the orthogonal matrix, the necessary and sufficient conditions of and the expression for the orthogonal solutions and the (skew-symmetric orthogonal solutions of the system of real matrix equations AX=B, XC=D are respectively derived for the first time. Then, combining the spectral theory of matrix, the (skew-)symmetric orthogonal least squares solutions are shown.In Chapter5, the symmetric arrowhead least squares solutions of the system of real matrix equations AXB=C, EXF=D and its optimal approximate solution are firstly discussed by using the special decomposition of the symmetric arrowhead matrix. Then, the symmetric arrowhead least squares solutions with a leading principal submatrix con-straint and its optimal approximate solution are respectively studied. As applications, the solvability conditions and the expressions of the symmetric arrowhead solutions and the symmetric arrowhead solutions with a leading principal submatrix constraint of the above system are respectively given.In Chapter6, we mainly summarize the dissertation, and meanwhile describe the problems that we can further study.