| The problem with a submatrix constraint is, under some constrained conditions,constructing a matrix A with a given matrix A0 as its submatrix. Note that the solutionset of the problem with a submatrix constraint is W(A0), and then the matrix inverseproblem with a submatrix constraint is, for given matrices X, B, finding a matrixA∈W(A0) such that AX = B . And the best approximate solution is the nearestmatrix to a given matrix in the mentioned matrix set above.The matrix inverse problem with a submatrix constraint has been widely used inmany fields such as structural design, system identification, dynamics, control theory,vibration theory and so on. We will study the best approximate solutions of the matrixinverse problems with some submatrixes constraint in this thesis. The main problemsdiscussed are as follow:Problem I Given X ,B∈Rn×m, A0∈S , letfind (A|)∈Γ, such thatwhere S ,S' are given sets,‖·‖is Frobenius norm.Problem II Given A*∈S'' , find (A|^)∈SE , such thatwhere SE is the solution set of Problem I.The solution of problem II is called the best approximate solution of the matrixinverse problem with a submatrix constraint in this thesis.In this thesis, the problems of finding the least-squares and the best approximatesolutions of the matrix equation AX = B with some submatrixes constraint are studiedwhen the set S is (anti-)centrosymmetric or symmetric and skew anti-symmetric oranti-symmetric and presymmetric or anti-bisymmetric set. By using the characterof Frobenius norm, the least-squares problems with some submatrixes constraint aretransformed into other least-squares problems with no constraint to discuss. Then byusing the generalized singular value decomposition and canonical correlation decompo- sition, the general expressions of the least-squares solutions are obtained respectively.In addition, the best approximation problems are also considered. The best approxi-mation solutions, numerical algorithms and examples are given.
|
PDF Full Text Request