Abstract: The following inverse problem is discussed: Problem DD. Given a n×n Jacobi matrix J_n and a set of distinct real numbers λ₁,λ₂,…,λ_n, Construct a 2n×2n Jacobi matrix J_2n whose eigenvalues are {λ_i}_i-1²ⁿ and whose leading n×n principal submatrix is J_n. The necessary and sufficient condition for the problem DD to have a solution is derived, and the algebraic expression of the solution is given if the solution exists. An algorithm of solving the problem DD is established on the basis of these results.

Key words: inverse problem, eigenvalue, minimal polynomial