| The constrained matrix equation problem has been widely used in control the-ory , civil engineering ,vibration theory ,economy field ,engineering calculationand so on.This paper studies the following questions of the least-squares and theoptimal approximation solution of the generalized re?exive matrices and general-ized anti-re?exive matrices :Problem1 : Given matrixX,B, find A in S such that AX = B, where S is thegiven set of matrix.Problem2 : Given matrixX,B, find A in S such that ||AX - B|| = min, where Sis the given set of matrix.Problem3 :Given matrix M,find N in SEsuch that ||M-N||= inf|| M -A|| ,whereSE is the solution set of Problem 1or Problem2 , ||.|| is the Frobenius norm.The main results of this paper are as follows:1.When S is the set of generalized re?exive matrices and generalized anti-reflexive matrices ,we have obtained the solvability of the sufficient and necessaryconditions of Problem 1,and the expressions of solutions of Problems 2,3 by usingthe character and construction of two kinds of matrices .At the same time ,we havealso given the numerical example to support our theoretical results.2.When S is the set of reflexive matrices and anti-reflexive matrices , on thelinear manifold,by applying the character and construction of two kinds of matri-ces ,we have discussed the Problems 2,3,and obtained the expressions of solutionsof Problem2 and the optimal approximation of Problem 3. At the same time ,the problems of matrix equation AX + XA = Chaving nonempty solution setin the generalized reflexive matrices or the generalized anti-re?exive matrices areconsidered ,and their general expression are provided.3.When S is the set of generalized reflexive matrices and generalized anti-reflexive matrices , on the linear manifold,we have obtained the expressions ofsolutions of Problem2 and the optimal approximation of Problem 3.The thesis is supported by the Nation Natural Science Foundation of China.
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