Several Classes Of Submatrix Extension Problems And A Class Of Constrained Matrix Equation Problems

The submatrix extension problem is to construct a matrix A with a given matrix or several given matrices as its submatrixs under some constrained conditions. The constrained matrix equation problem is to find a solution or a least-sequare solution in a constrained matrix set. The matrix extension problems and constrained problems of reflexive matrix or anti-reflexitive matrix come from many fields and have been widely used in science computation, structural design, systemic parameter identification, control theory, quantum mechanics, electricity, vibration analysis, nonlinear theory, dynamic program and so on.The main results of this thesis are as follows.Firstly, the submatrix extension problem is discussed in the second chapter. In this paper, it is mainly considered that the real submatrix A₀ extends the real matrix A under the constrained conditions //AX -B// = min, that the two real submatrices A_o, A₁ extends the real matrix A under the constrained conditions ||AX-B|| - min, that the symmetric submatri A₀ extends the real symmetric matrix under the constrained conditions AX - B, and that the symmetric submatrix A₀ and the skew antisymmerix A-1 extends the bisymmetric matrix under the constrained conditions AX = B. Moreover, the conditions of the solution set being nonempty set and the expressions of the solutions of the above extension problems are derived. The optimal approximation solutions of the solution set and a given matrix A^* is also considered, the numerical algorithms and numerical examples are given.Secondly, the constrained matrix equation problems of the reflexive matrix or anti-reflexive matrix is discussed in the third chapter. Their least-sequares solutions, their optimal approximation on linear manifolds and the necessary and sufficient conditions for the equation AX = B,XC = D having nonempty solution set in the reflexive matrix or the anti-reflexive matrix set are respectively considered. Moreover, when solution sets of the above problems are nonempty, their general expressions are provided. The optimal approximation solution of the solution set and a given matrix A^* is obtained.