Research For Several Constrained Matrix Equations And The Optimal Approximation Based On The Iterative Method

Constrained matrix equation problem is to find the solutions of a matrix equation (group) which satisfies a few constraint conditions. As a leading direction in the field of modern numerical algebra, Constrained matrix equation problem and the extension of the new problem is widely applied in structural design, control theory, vibration theory, cycle theory, thermal system, parameter identification, electrical, the linear optimal control and other fields.This paper mainly discusses the following problems:Problem Ⅰ Given A∈Cmxn, B∈Cmxx, S∈Cnxn, find X∈S, such that AXAH=B.Problem Ⅱ Given A, B∈Cnxn, find X∈S(?)Cnxn, such that AX＝B.Problem Ⅲ When problem I or II is consistent, let SE denote the set of its solutions. For given X∈Cnxn, find X∈SE such thatWhere (?) is Frobenius norm, S is the set of the Hermite matrix or the anti-Hermite matrix or the row symmetric matrices or the row anti-symmetric matrix.When S is the set of Hermite matrix or anti-Hermite matrix or the row symmetric matrix or the row anti-symmetric matrix respectively, the orthogonal projection iterative solution of problem Ⅰ and Ⅲ, problem Ⅱ and Ⅲ is discussed. Firstly, the orthogonal projection iterative algorithms is constructed by using the structure and properties of the (anti-) Hermite matrix and the (anti-) symmetric matrix; Additionally, the convergence of algorithm is proved by using singular value decomposition of the matrix and the orthogonal invariance of the F-norm.; And the convergence rate estimate of the algorithm is given; Finally it has been confirmed that the algorithm is efficient and feasible by the numerical experiments.