Abstract: This paper considers the following two inverse eigenvalue problems:Problem Ⅰ. Given λ,μ∈R, λ > μ, and x, y∈Rⁿ:x≠0,y≠0, x^Ty=0. Find n×n Jacobian matrix J such that Jx=λx, Jy=μy; λ > μ > λ₃(J) > … > λ_n(J)(or λ₁(J) > … > λ_n-2(J) > λ > μ).problem Ⅱ. Given x∈Rⁿ, x≠0, and n distinct real numbers λ₁,λ₂,…,λ_n which satisfy λ₁ > λ₂ > … > λ_n. Find n×n Jacobian matrix J such thatλ_i(J)=λ_i, i=1,…,n; Jx=λ₁x (or Jx=λ_nx)Some necessary and sufficient conditions for existance of solution of these problems are given. For the problem Ⅰ, the expression of the solution is given. For the problem Ⅱ, a numerical algorithm is provided.

Key words: inverse eigenvalue problem, nonnegative matrix, characteristic vector, similar transformation