Some Research For Matrix Equation Problems Under Submatrix Constraint

The constrained matrix equation problem is to find the solution of a matrix equation in a constrained matrix set. The study of it has been a hot topic in the field of numerical algebra in recent years. Actually, it is widely used in many fields such as structural design, system identification, structural dynamics, automatics control theory, vibration theory. When a submatrix constranit is one of the con-stranied conditions of the matrix set, a constrained matrix equation problem then is called a matrix equation problem with a submatrix constranit, namely, to find a matrix A as the solution of a constrained matrix equation with a given matrix A₀ as its submatrix. The main works and results of this dissertation are as follows.1. The problem of finding the symmetric nonnegative definite solutions of matrix equation AX = B with a submatrix constranit is studied. The necessary and sufficient conditions for the existence of and the general expression for the solutions are obtained. Furthermore, the results for the relative inverse eigenvalue problem are also obtained. The problem of finding the symmetric and anti-symmetric least-squares solutions of matrix equation AX = B with a submatrix constranit is discussed. By using the subspace theory and the Projection Theorem, the least-squares problems are transformed into equation problems to discuss. Finally, The general expressions of the symmetric and anti-symmetric least-squares solutions are obtained respectively. In addition, the optimal approximation problems are also considered. The optimal approximation solutions, numerical algorithms and examples are given.2. The Hermitian-Hamiltonian, Hermitian Anti-Hamiltonian, Anti-Hermitian Hamiltonian and Anti-Hermitian Anti-Hamiltonian solutions of matrix equation AX = B with a submatrix constranit are considered. By the analysis of the structrual characterizations of these matrices and the singular value decomposition, generalized singular value decompositions of matrix, the necessary and sufficient conditions for the existence of and the general expressions for the solutions of these problems are given. Whats more, The relative optimal approximation problems are discussed, and the optimal approximation solutions are obtained.3. To those matrices mentioned in 2, their least-squares solutions of matrix equation AX = B with a submatrix constranit are studied. First the least-squares problems are transformed into equation problems using the method in 1. Then by using the structrual characterizations of these matrices, the singular value decom-...