Types Of Constraint Matrix Equation Problem

The constrained matrix equation problems have been widely used in many fields such as control theory, vibration theory, system parameters identification, nonlinear programming and so on. In this paper, we will systematically study several kinds of constrained matrix equation problems from different aspects: recursive algorithm and direct calculation method by using singular value decomposition, the canonical singular value decomposition and the generalized singular value decomposition of matrices. The main problems discussed are as follows:Problem Ⅰ Given X,B∈ R^n×m and the matrices set S (?) R^n×n, find A∈S such that AX = B.Problem Ⅱ Given X, B ∈ R^n×m and the matrices set S (?) R^n×n, find A ∈ Ssuch that ||AX - B|| = min .Problem Ⅲ Given A ∈ R^n×m, B ∈ R^m×m and the matrices set S (?) R^n×n, find X∈S such that||A^TXA-B|| = min.Problem Ⅳ Given (?) ∈ R^n×n , find A^* ∈ S_E such thatwhere S_E is the solution set of Problem Ⅰ , Ⅱ or Ⅲ and ||·|| is the Frobeniusnorm.The main results of this paper are as follows:1. A recursive algorithm to solve matrix equation AX + XB = F over symmetric solutions is constructed. By this algorithm, the solvability of the equation over symmetric solutions can be determined. When the matrix equation is consistent, the symmetric solutions can be obtained and its least-norm symmetric solution can be given by choosing a special initial matrix. The recursive algorithm to solve matrix equation AXB = C over anti-symmetric solutions is also discussed.2. We present the sufficient and necessary conditions for Problem I and give the expressions of solutions for Problem Ⅰ and Problem Ⅳ when S is the set of all symmetric orth-symmetric matrices. The expressions of the solutions for Problem Ⅱ and the corresponding problem Ⅳ are get, in which S can be written asS = {A ∈ SAR_pⁿ| ||AZ - Y|| = min}. In addition, by using the canonical singular value decomposition, the expressions of solutions for Problem Ⅲ is solved.3. Over the linear manifold S = {A ∈ AAR_pⁿ| AZ = Y,Y_i^TZ_i = -Z_i^TY_i,Y_iZ_i⁺Z_i =Yt,i = 1,2}, the expressions of solutions for Problem II and the correspondingproblem IV are given. When S is the set of all anti-symmetric orth-aniti-symmetric matrices, the expressions of solutions for Problem III is derived by the canonical singular value decomposition of matrices.4. The sufficient and necessary conditions for Problem I and the expressions of solutions for Problem I and Problem IV are given when S is the set of all anti-symmetric orth-symmetric matrices. By applying the generalized singular value decomposition of matrices, we derive the necessary and sufficient conditions for the existence of the anti-symmetric orth-symmetric solution of linear matrix equation AJXA = B . The general expression of the solutions for A^XA-B and its corresponding problem IV are given. In addition, the expressions of the solutions for Problem II and the corresponding problem IV are provided over the set S whichcan be written as S = \A e ASR'^, AZ - F|j =...