Jacobi矩阵的逆特征值问题

计算物理 ›› 1992, Vol. 9 ›› Issue (3): 313-322.

Jacobi矩阵的逆特征值问题

吕烔兴, 耿长宏

南京航空学院, 数理力学系, 210016

收稿日期:1991-10-01 修回日期:1992-03-30 出版日期:1992-09-25 发布日期:1992-09-25

INVERSE EIGENVALUE PROBLEMS FOR JACOBIAN MATRICES

Lü Tongxing, Geng Changhong

Nanjing Aeronautical Institute, 210016

Received:1991-10-01 Revised:1992-03-30 Online:1992-09-25 Published:1992-09-25

摘要/Abstract

摘要： 本文研究以下两类逆特征值问题:问题1:给定实数λ,μ(λ > μ)和n维非零正交实向量x,y。求n阶Jacobi矩阵J,使得Jx=λx,Jy=μy,λ > μ > λ₃(J) > …λ_n(J)(或λ₁(J) > … > λ_n-2(J) > λ > μ)。问题2:给定n维非零实向量x和n个实数λ₁ > … > λ_n。求n阶Jacobi矩阵J,使得λ_i(J)=λ_i,i=1,…,n,Jx=λ₁x(或Jx=λ_nx)。文中导出了问题有解的一些充分必要条件。对问题Ⅰ,给出了解的表示式:对问题Ⅱ,提供了一种数值解法。

关键词: 逆特征值问题, 非负矩阵, 特征向量, 相似变换

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

吕烔兴, 耿长宏. Jacobi矩阵的逆特征值问题[J]. 计算物理, 1992, 9(3): 313-322.

Lü Tongxing, Geng Changhong. INVERSE EIGENVALUE PROBLEMS FOR JACOBIAN MATRICES[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 1992, 9(3): 313-322.

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