Abstract: In this paper we consider the equation AnX =λCnX ,where Anis a symmetric tridiagonal matrix and Cnis a diagonal matrix.Regarding Anas a 3×3blocked matrix,given a(r+1)×(r+s)non-sequential principle submatrix of An ,given Cn ,four vectors X1 = (x1,…,xr)＇,X3=(xr+s+1,…,xn)＇,Y1 = (y1,…,yr)＇,Y3=(yr+s+1,…,yn)＇and two distinct real numbersλ,μ,we construct a symmetric tridiagonal matrix Anand two vectors X2 = (xr+1,…,xr+s)＇,Y2= (yr+1,…,yr+s)＇such that AnX =λCn X and AnY =μCn Y ,where X = (X1＇, X2＇,X3＇)＇,Y = (Y1＇,Y2＇,Y3＇)＇.The existence conditions such that the problem has a solution and the corresponding algorithm to find the solutions are given.A numerical example is presented to show the validity of the algorithm.

Key words: Symmetric tridiagonal matrix, Diagonal matrix, The inverse generalized eigenvalue problem,  , Nonleading principle submatrix, Defective generalized eigenpair