Several Classes Of Matrix Extension Problems And Two Kinds Of Constainted Matrices Equations

The problem of matrix extension is, under some constrained condition, to construct matrices A with a given matrix A0 as its submatrix;The problem of the constrained matrix equation is to find the solution of matrix equation in some matrices set, the following problems are considered.Problem I Given X , B∈R n×m , A0∈ASR q×q ( R q×q),1≤q≤n, find A∈ASRPn×n( ACSRn×n), such thatProblem II Given X , B∈R n×m; A0∈SRq×q(ASRq×q,Rq×q),1≤q≤n,find A∈S1, such that where S1 is the solution set of f ( A)= AX?B=min, A∈SRn×n( ASR n×n,D2 SRn×n).Problem III Given A~∈Rn×n,find A?∈S,such that where S is the solution set of Problem I or II.Problem IV Given X∈Rm×k, Y∈Rn×l, C∈Rm×l, B∈Rn×k,find A∈RPn×Qm ( R?nP×mQ), such thatProblem V Given A~∈Rn×m, find A?∈S,such that where where S is the solution set of Problem IV.By applying the SVD,GSVD,QSVD,the soluable sufficient and necessary conditions and the general solution of Problem I,II are given, the optimal approximate solution of Problem III is finded;By using the character of the real general reflexive and anti-reflexive matrices,the general solution of problem IV and approximate solution of ProblemV are obtained, the numeral algorithms and the examples of optimal approximation solution are given.