Constrained Solutions For Matrix Equations And Their Opitimal Approximations

The constrained matrix equation problems have been widely used in many fields such as structural analysis, control theory, vibration theory, nonlinear programming and so on. The research works on constrained matrix equation problems have significent theoretical and practical value. In this paper, we study the following problems:Problem I Given A,C∈R^m×n,B,D∈ R^n×m,S (?)R^n×n. Find X∈ S, such that(AX,XB) = (C,D);Problem II Given A,C∈R^m×n,B,D∈R^n×m,S(?)R^n×n. Find XeS ,such thatProblem III Given A∈R^m×n,B∈R^n×s,C∈R^m×s,S(?) R^n×n Find X∈S, such thatAXB = C;Problem IV Given A∈R^m×n,B ∈ R^n×s,C∈R^m×s,S (?)R^n×n. Find X∈ S, such thatProblem V Given X^*∈R^n×n Find X∈S_E, such thatwhere S_E denotes the solution set of Problem I or Problem II or Problem III orProblem IV; ||·|| is the Frobenius norm.The main results of this paper are as follows:1.We establish the necessary and sufficient condition for the existence of a solution of Problem I and give the expressions of solutions for Problem I and Problem V where S is the set of anticentrosymmetric matrices.2.Over the linear manifolds = {X ∈ ACSR^n×n| ||A₀X-B₀||= min, A₀,B_o∈ R^m×n} ,wesuccessfully solve Problem II and Problem V; and give the expressions of solutions for Problem II and Problem V ;and provide a numerical method of finding the...