The Generalized Inverse Eigenvalue Problems Of Row (Column) Symmetric Matrices And Their Optimal Approximation

The generalized inverse eigenvalue problems and optimal approximation problemsof matrices are one of the most important study fields of the numerical linear algebra.They have been widely applied in a variety of fields and play an important part in emen-dation of finite element dynamic model and the recovery or correction of linear systems,such as solid mechanics,molecular spectroscopy,vibration control,parameters of systemidentification,electricity,automatic control,structural vibration.In this thesis,the following generalized inverse eigenvalue problems and optimalapproximation of row(column) symmetric matrices and row(column) anti-symmetricmatrices are studied.Problem I. Given X∈Rn×k,Λ= diag(Λ1,···,Λl)∈Rk×k, whereΛi(i =1, 2,···, l) is real matrix of one order or two orders and sets S A ? Rn×n, S B ? Rn×n,findA∈S A, B∈S B, such thatAX = BXΛ.Problem II. Given A?∈Rn×n, B?∈Rn×n, find ( A?, B?)∈S AB, such that( A?, B?) ? ( A?, B?) F = ?(Ai,Bn)f∈SAB (A, B) ? ( A?, B?) F,where S AB is the solution set of Problem I ,i.e.S AB = { (A, B) | AX = BXΛ, A∈S A, B∈S B }.The problems mentioned above are discussed at the situation as follows:1. S A and S B are the sets of row symmetric matrices;2. S A and S B are the sets of row anti-symmetric matrices;3. S A is the set of row symmetric matrices, S B is the set of row anti-symmetricmatrices;4. S A is the set of row anti-symmetric matrices, S B is the set of row symmetricmatrices;5. S A and S B are the sets of column symmetric matrices;6. S A and S B are the sets of column anti-symmetric matrices;7. S A is set of column symmetric matrices, S B is set of column anti-symmetricmatrices;8. S A is set of column anti-symmetric matrices, S B is set of column symmetricmatrices. On the base of studying the basic properties of the row(column) symmetric ma-trices and row(column) anti-symmetric matrices, the expressions of the general solu-tions to problem I, the expressions of optimal approximation solutions, together withthe corresponding algorithms and numerical examples are presented at each situationmentioned above.