Hermitian And Generalized Hamiltonian Constraint Matrix Equation

The constrained matrix equation problems have been widely used in control theory, vibration theory, computational physics and nonlinear programming. This doctoral dissertation studies systematically several types of constrained matrix equations, we discuss primarily the problems as follows:Problem I Find A S , such thatProblem II Given B Cnxm , ScCn×n. Find A e S such thatAX = B. Problemlll Given X, 5 e Cnxm , ScCn×n. Find A S such that|AX-B|| = min.Problem IV Given .4* Cn×n . Find A e S, such thatHere SB is the solution set of problem I, II or III, and || - | is Frobenius norm. S is respectively any one of these four types of matrices - HHCn×n, HAHC? AHHCn×n and AHAHCn×nThe main research results in this dissertation are as follows:1. The dissertation discusses respectively the property and structure of the matrix classes- HHCn×n HAHCn×n, AHHCn×n and AHAHCn×n, especially the characteristicproperties in these matrix sets. The space decompounding theorem and the denotative theorem of the matrix are achieved by using these properties.2. By using the properties of the eigenvalue and eigenvector of the several kinds of matrices, this dissertation presents subtly and reasonably the mathematical description of inverse eigenvalue problem (Problem I). By using the properties of these matrices, this dissertation obtains the sufficient and necessary conditions of the solvability, the solution structure and expression of the general solution for Problems I and II, when S is chosen HHCn×n HAHCn×n, AHHCn×n and AHAHCn×n, respectively. Byusing the denotative theorem of these matrices and SVD decompounding method, the sufficient and necessary condition of the solvability and the expression of the general solution are obtained for Problem III.3. As for the relevant Problem IV, the dissertation attests the existence and uniqueness of the optimal approximation solution and presents the expression of the optimal approximation solution.4. With regards to the matrices mentioned above, this dissertation studies further the least-squares problem and the optimal approximation problem on the linear manifold, attests the existence and uniquence of the optimal approximation solution, and presents the expressions of the least-squares solution and the optimal approximation solution. In addition, Using the denotative theorem of the matrix in the set of HAHC"*", this dissertation presents sufficient and necessary condition that positive semidefinite solutions exist and the expression of the general solution for Problem II, and gives the expression of the only optimal approximation solution for the relevant Problem IV. At the end of this dissertation, we present some numerical test results, which support our theoretical results.The dissertation has gained the support from the National Natural Science Foundation of China.