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.

This paper considers the problem of constructing a Jacobi matrix from its defec-tive eigen-pair and a principal submatrix. Some necessary and sufficient conditionsof solvability have been derived. An algorithm and two numerical examples havebeen given.

给定矩阵X和B，得到了矩阵方程XTAX=B有双对称解的充分必要条件及有解时解的一般表达式。用SE表示此矩阵方程的解集合，证明了SE中存在唯一的矩阵A，使得A与给定矩阵A*的差的nbenius范数最小，并且给出了矩阵A的表达式。

本文研究了两个Jacobi矩阵的逆特征问题：I给定实数λ，μ(λ>μ)和n维非零实向量x1，y1求n阶Jacobi矩阵J1使Jz=λx，Jy=μy，且λ>λ2(J)>…>λ2-1(J)>，λ>λ1+1(J)>…>λn(J)，或λl(J)>λ2(J)>…λ1-1(J)>λ>λi+l(J)>…>λn-1(J)>μ。Ⅱ给定实数λ1μ(λ>μ)和n维非零实向量x1y1求n阶Jacobi矩阵J，使Jx=Az=λy=Jy，且λ1(J)>λ2(J)>…>λ2(J)>λ>λ>μ>λ1+2(J)>…>An(J)文中给出了问题Ⅰ，Ⅱ有唯一解的充要条件，并给出了解的表达式。