The Related Matrix Equations Problems Of Row (Column) Symmetric Matrices And Their Optimal Approximation

The constrained matrix equation problem is to find a solution of the matrix equa-tion in a constrained matrix set. This problem has a wide ranges of applications inmany areas, especially in the structural design, system identification, principal com-ponent analysis, exploration and remote sensing, biology, electricity, solid mechanics,molecular spectroscopy, structural dynamics, automatics control theory, vibration the-ory, circuit theory, and so on. The results of the study have important background ofapplication.In this thesis, the following linear constrained matrix equations problems and theleft and right inverse eigenpairs problems of row(column) symmetric matrices are stud-ied. In addition, the corresponding optimal approximation problems are also discussed.Problem I. Given A∈Rh×n, B∈Rh×m, C∈Rm×l, D∈Rn×l,and a special matrixset S∈Rn×m, find X∈S , such thatProblem II. Given X∈Rn×h,Λ= diag(λ1,···,λh)∈Rh×h, Y∈Rn×l,μ=diag(μ1,···,μl)∈Rl×l, and a special matrix set S∈Rn×n,find A∈S ,such thatProblem III. Given X?∈Rn×m, find X?∈S E, such thatwhere S is a set of row symmetric(antisymmetric) matrices or a set of column sym-metric(antisymmetric) matrices, S E is the solution set of Problem I or II ,·is theFrobenius norm.First of all, the properties of row symmetric(antisymmetric) matrices and columnsymmetric(antisymmetric) matrices, and their eigenpairs are discussed. Secondly, basedon singular value decomposition of the matrix, the su?cient and necessary conditionsof Problem I and II , the expression of the solutions to Problem I and II and the expres-sion of the optimal approximation solution to the corresponding Problem are obtainedrespectively. The numerical algorithm and the numerical examples to find the optimalapproximation solution are given.