Fixed Rank Solutions Of Several Classes Of Constrained Matrix Equations

The constrained matrix equation have been widely applied in system engineering, automatic control theory, economics, network planning, civil engineering and vibration theory. Solving for fixed rank solutions of constrained matrix equation has great significance to perfect the theory of constrained matrix equation. The following problems are considered systematically in this M.S. thesis:Problem I Givendetermine the maximal and minimal ranks of X in S1 , and give the representations of the elements inProblem II Givendetermine the maximal and minimal ranks of X in S 2, and give the representations of the elements inProblem III Given determine the maximal and minimal ranks of X, Y in S 3, and give the representations of X and Y , when the ranks reached their minimal, respectively. determine the maximal and minimal ranks of X and Y in S 4, respectively.The main achievements are as follows:1. For problem I, when the set S are cetro-symmetric, symmetric and bisymmetric matrices, respectively, the maximal and minimal ranks of X in S1 and the representations of the elements in S1 * are obtained, by using singular value decomposition (SVD) and quotient singular value decomposition (QSVD). For S = CSRn×n in particular, the optimal approximate solutions to a given matrix are also given, as well as the algorithm and the numerical example.2. For problem II, when B = A and S = R^n×n or S = SR^n×n, the fixed rank solutions are derived by using SVD; when B is a general real matrix, problem II is solved mainly using QSVD. 3. By applying restricted singular value decomposition (RSVD) and the rank inequalities, problem III is solved.4. problem IV is solved by using SVD and the theories on ranks. In addition, the problem of ranks of bisymmetric solutions to the matrix equation AXA^T = C is changed into problem IV, and the relevant results are obtained.