| The linear matrix equation is widely appeared in many fields such as structural analysis, system parameters identification, automatic control, nonlinear programming and so on. The research works on matrix equations have significant theoretical and practical value. In this paper, we study the following problems:Problem I Given , such thatProblem II GivenX,Bî–˜nxm,S C Rttxn .Find^G5 , such thatProblemIII Given A Rm n,B Rn p,D Rm p,S Rn n.Find X S, such thatProblem IV Given A* Rn n .Find A SE, such thatwhere SE denotes the solution set of Problem I or n or III.The main results of this paper are as follows:1.We give the expressions of solutions for Problem I and Problem II, in which S is abounded closed set can be written as: S {A ORn n | || AY - Z ||= min} .In addition, thecorresponding problem IV is also solved, which generalize and deepen the results of Professor Zhang's.2.When S is the set of all symmetric orthogonal nonnegative definite matrices, Problem I and Problem IV are discussed and solutions for the two problems are derived.3.Over the linear manifoldS {A ASRPn n | ||AZ-Y||=min},we successfully solve Problem II and Problem IV. When S is a subspace, which is defined as: we present the necessary and sufficientconditions for Problem I and give expressions of solutions for Problem I and Problem IV, too. In fact, Problem I becomes inverse eigenvalue problem if B is replaced by X A ,Where A is a diagonal matrix, the Similar Re-nonnegative definite and Similar Re-positive definite solutions for Problem I are given. 4.By applying QSVD and GSVD, We separately get the expressions of solutions forProblem III when S is a linear manifold S {X ACSRn n| ||XZ-Y||= min} or S isthe set of Hermite matrices. Apparently, matrix equation AXAT = B can be considered as a special case of AXB = C, the least-squares symmetric orthogonal symmetric solutions for it are obtained over a linear manifoldS = {X SPRnxn | || XZ - Y ||= min} by using CCD of matrix pairs.
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