完成具有特定最小最大特征值的Jacobi矩阵

摘要

通过使用Jacobi矩阵的各阶顺序主子阵特征多项式之间的递推关系，解决了具有特定最小最大特征值的Jacobi矩阵完成问题。证明了这个矩阵完成问题存在唯一解的充要条件，得到了被插入元素的递推表达式。然后利用这些结果解决了具有2n−1个极端特征值的Jacobi矩阵的逆特征值问题。最后，提出了确定Jacobi矩阵的相应算法。数值实验验证了这些算法的有效性。%By using the recurrence relation between the characteristic polynomials of the lead-ing principal submatrices of a Jacobi matrix, we resolve the Jacobi matrix comple-tion problem with prescribed minimal and maximal eigenvalues. The necessary and sufficient condition for that the problem exists the unique solution is proved. More-over, the recurrence relations of the inserted elements are obtained. Furthermore, based on these results, the inverse eigenvalue problem of the Jacobi matrix with 2n−1 extreme eigenvalues is solved. Finally, two algorithms for the problem are proposed. Numerical examples verify the effectiveness of the proposed algorithms.