Let P ∈ Rn×n such that pT=p, p-1=pT. A ∈ Rn×n is termed symmetric orthogonal symmetric matrix ifAT=A, (PA)T=PA.We denote the set of all n×n symmetric orthogonal symmetric matrices by SRn/p×nThis paper discuss the following two problems:Problem I. Given X Rnxm, A=diag(λ1,λ2...,λm). Find A ∈SRn/p×nsuch thatAX = XA.Problem Ⅱ. Given A ∈Rn×n. Find A* ∈SE such that‖A-A*‖=in/A∈SE‖A-A*‖where SE is the solution set of Problem I, ‖A-A*‖is the Frobenius norm In this paper, the sufficient and necessary conditions under which SE is nonemptyare obtained. The general form of SE has been given. The expression of the solu-tion A* of Problem II is presented. We have proved that some results of Reference[3] are the special cases of this paper.