Jacobi矩阵的逆特征问题

INVERSE EIGENPROBLEMS FOR JACOBIAN MATRICES

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x，y，求n阶Jacobi矩阵J,使Jx=λx，Jy=μy,且λ＞λ2（J）＞…＞λi-1（J）＞μ＞λi＋1（J）…＞λn（J），或λi（J）＞λ2（J）＞…＞λi－1（J）＞λ＞λi＋1（J）＞…＞λn-1（J）＞μ·II给定实数λ，μ（λ＞μ）和n维非零实向量x，y，求n阶Jacobi矩阵J，使Jx＝λx，Jy＝μy，且λ1（J）＞λ2（J）＞…＞λi－1（J）＞λ＞μ＞λi＋2（J）＞…＞λn（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．

This paper considers the following two inverse eigenproblems:(I) Given λ,μ∈R (λ >μ), and 0 ≠x, 0≠ y∈ Rn. Find an n × n Jacobian matrix J such that Jx =λx, Jy = μy, where (a) λ> λ2(J) >... > λi-1(J) >μ > λi+1(J) >... > An(J),(b) λ1(J) >... > λi-1(J) > λ> λi+1(J) >... >λn-1(J) > μ(II) Given λ,μ∈R (λ>μ), and 0 ≠x, 0 ≠y ∈Rn. Find an n ×n Jacobian matrix J such that Jx =λx,Jy =μy, where λ1(J) >λ2(J) >... >λi-1(J) > λ> μ >λi+2(J) > ... >An (J).Some necessary and sufficient conditions for the existence of unique solution to these problems are given. Also explicit expression of the solution are given.