Study On Solutions Of Matrix Equations With Submatrix Constraints

Matrix equation problems with submatrix constraints are derived from practical problems such as system theory,control theory and stability analysis,which have wide application background.These matrix equation problems have attracted the attention of scholars and become a hot topic,and their research has achieved important results.Many methods have been constructed to solve the matrix equation problem under submatrix constraints,such as conjugate gradient method,matrix decomposition method,parameter iteration method,etc.In this paper,the conjugate gradient iteration method is mainly used to solve the solution of the matrix equation under submatrix constraints and its optimal approximation.By using the properties and structural characteristics of special matrices,an iterative algorithm satisfying certain submatrix constraints is constructed.Finally,a numerical example is given and the effectiveness of the algorithm is proved by Matlab.The research contents are as follows:1.The generalized Hamiltonian solution and its optimal approximation of the matrix equation AXB+CXD=F under the sub matrix constraint are studied.The iterative algorithm of the matrix equation under the submatrix constraint is constructed by using the conjugate gradient method.The correctness of the results and the effectiveness of the algorithm are verified by using the Matlab mathematical programming language.2.The generalized inverse Hamiltonian solutions of linear operator equations φ1(X)=C1,φ2(X)=C2 are solved under subspace constraints.The equations A1XB1=C1,A2XB2=C2 are taken as an example to discuss,and the iterative algorithm for the solution and its optimal approximation under generalized inverse Hamiltonian matrix constraints is constructed.Finally,the feasibility of the algorithm is verified by numerical examples.3.In the complex domain,the Hermite anti-reflexive solution of the generalized Sylvestre matrix equation ∑j=1lAijXjBij=Fi,i=1,2,…,s with respect to matrix P under the constraint of submatrix is discussed.By using the conjugate gradient iterative algorithm and the structural characteristics of Hermite matrix,the conjugate gradient iterative algorithm satisfying the constraint of submatrix in the complex domain is constructed.A numerical example is given to verify the correctness and feasibility of the algorithm.