An Iterative Method And Optimal Approximation Solution Of The Constrained Matrix Equation AX = B

The constrained matrix equation problem plays an important role in structural de-sign.system identification,automatics control theory,vibration theory and other fields, its study has been one of the topics of very active research in the field of numerical algebra in recent years. Hitherto we have achieved many results in these fields.The problems discussed in the M.S thesis are as follows:Problem 1.Given A, B∈Cm×n,S C Cn×n,find X∈S,such that AX=B.Problem 2. Given X0∈Cn×n,find X∈SE,such thatWhere‖‖F is Frobenius norm,SE is the solution set of Problem 1.In this paper, matrix equation AX=B and the related optimal approximation are studied in the set of the complex matrix systematically,such as the set of Hermitian matrix, anti-Hermitian matrix, reflexive matrix, anti-reflexive matrix, Hermitian reflex-ive matrix,Hermitian anti-reflexive matrix.Firstly, based on the properties and con-struction of these kinds of matrix and by using the orthogonal projection,the iterative method is constructed.Secondly by using singular value decompositions,the orthogonal projection,the invariant of orthogonal transformation of Frobenius norm, the conver-gence of the method is proved.Furthermore the related optimal approximation can also be obtained with the method which only need to be made slight changes.Thirdly the convergence rate is analyzed and the estimation of the convergence rate is given.Finally the numerical examples are given,so the effectiveness is show.The M.S thesis is supported by Nature science Foundation of China(10671072).