Study On The Algorithms Of Generalized Symmetric Constraint Matrix Equation And Its Least-Squares Problems

The constrained matrix problem is widely used in the fields of financial engineering,systems engineering,image restoration and cybernetics.Many experts and scholars have paid attention to these problems and obtained a series of remarkable achievements.The generalized Constraint matrix is an extension of the constraint matrix,which has wider application range and more diversified problems to be solved,such as:multiplying a general non-symmetric matrix by a non-singular real matrix,making the new product a symmetric matrix;The positive definite matrix is multiplied by the General matrix to get the symmetric matrix,which can be used to solve the problems in probability theory.This kind of problem has great application value to engineering technology.In this thesis,we study a class of generalized constrained matrix equations which multiply an arbitrary matrix by a primitive matrix to make its product a symmetric matrix.The main research work is as follows:Problem ?.Given matrices M ? Rm×n,A ? Rp×n,B ? Rp×m,Find X ? MSRnxn,such that AX=B,min?AX-B?F2Problem ?.Given matrices M ? Rm×n,A ? Rp×n,B ? Rp×m,Find X ? MASRnxn,such that AX=B,min?AX-B?F2Problem ?.Given matrices M ? Rm×n,A ? Rp×n,B? Rp×m,Find X ? MSR0n×n,such that#12Based on the matrix singular value decomposition,the generalized inverse of matrix and the matrix partition method,the general expressions of the necessity and sufficiency and the solutions of problems ? and ? are given,by using the generalized singular value decomposition of matrix,the optimal approximation solutions of problems ? and ? are obtained.The general expression of the solution of Problem ? is given by spectral decomposition of matrix.Numerical examples are used to illustrate the effectiveness of the algorithms.