The Least-squares Solutions Of Some Classes Matrix Equtions With A Submatrix Constraint

The constrained matrix equations have been widely used in many fields such asstructural analysis, control theory, vibration theory, nonlinear programming and soon. The research worked on constrained matrix equation problems has importanttheoretical and practical value.This paper is organized as follows.In the first chapter, the background, the necessity and the progress for thestudy of the constrained matrix equation problems are presented. Then, we intro-duce our main work simplely in the end.In the second chapter, the matrix equation XA = B with a submatrix con-straint is studied when the matrix X is someone matrix. At the same time, wegive the expressions of the least-squares solutions and the optimal approximationsolutions.In the third chapter, the matrix equation AXB = C with a submatrix con-straint is studied in this part. At the same time, we give the expressions of theleast-squares solution and the optimal approximation solution.In this paper, we study the following problems:Problemwhere S is a constrained matrix set with n×n. And X[1 : q] is a q×q leadingprincipal submatrix of the matrix X.Problem II: Given , such thatwhere S1 is the solution set of Problem I.Problem III: Given such thatProblem IV: Given where S3 is the solution set of Problem III.The main results of this paper are as follows:1. When S is the symmetric orthogonal symmetric matrix set, or the symmet-ric orthogonal anti-symmetric matrix set, or the anti-symmetric orthogonal sym-metric matrix set, or the anti-symmetric orthogonal anti-symmetric matrix set, wegive the expressions of the least-squares solutions and the optimal approximationsolutions of the matrix equation XA=B by using the generalized singular value de-composition(GSVD), the canonical decomposition(CCD) of the part matrices andthe project theorem, namely the solutions of the Problem I and the Problem II.2. We give the least-squares solution and the optimal approximation solution ofthe matrix equation AXB=C by using the generalized singular value decompositionand the canonical decomposition of the part matrix, and we give the su?cientprerequisite of the matrix equation, namely the solutions of the Problem III andthe Problem IV.3. We also give algorithms and examples of the problems II and problems IVnarrated above matrixs.