Types Of Constraint Matrix Equations And Its Optimal Approximation

The constrained matrix equation problems have been widely used in control theory, economic field, vibration theory, civil engineering and so on. We will systematically study several kinds of constrained matrix equation problems in this Ph.D. Thesis, the main problems discussed are as follow:Problem I GivenX e Cn x m, A = diag(λ1, λ2,.... λm), S Rn×n. Find A ∈ S such thatAX = X. Problem II Given X,B e Rn×m, S Rn×n. Find A ∈ S such thatAX = B. Problem III Given X,B ∈ Rn×m, S Rn×n. Find A ∈ S such that||AX -B|| = min.Problem IV Given A ∈ Rn×n. Find A' ∈ SE such that||A*-A||= min||A-~A||. A∈Sswhere SB is the solution set of Problem I or II or III The main results of this paper are as follows:1. When S is the set of all centrosymmetric matrices(nonnecessary bisymmetric), we have obtained the solvability conditions of Problem I , n and the expressions of solutions of ProblemI , II, III, IV by using the eigen-pair character and construction of this kind of matrices. Then we have also discussed the above problems respectively, when 5 is the set of all anti-centrosymmetric matrices.2. When 5 is the set of all symmetric and ortho-symmetric matrices, we have first discussed the construction and eigen-pair character of this kind of matrics. Then, by applying the eigen-pair character , the orthogonal projection and the general form of this kind of matrices, we have studied Problem I , II , in and IV, and obtained the solvability conditions of Problem I s n and the expressions of solutions of Problem I , II, III, IV. We have also solved the related problems.3. When 5 is the set of all symmetric and ortho-antisymmetric matrices, we have discussed the construction of this kind of matrices, and obtained the solvability conditions of Problem I and the expressions of solutions of Problem III and IV.4. When 5 is the set of all anti-symmetric and ortho-symmetric matrices, we have discussed the construction of this kind of matrices, and obtained the expressions of solutions of Problem III and IV.5. When 5 is the set of all anti-symmetric and ortho-antisymmetric matrices, we have first discussed the construction and eigen-pair character of this kind of matrices, then obtained the solvability conditions of Problem I and the expressions of solutions of Problem I and IV.This Ph.D.Thesis is supported by the National Natural Science Foundation of China. This Ph.D.Thesis is typeset by CLATEX .