箭头矩阵在最小最大特征值条件下的完成问题

Completing arrowhead matrices with minimal and maximal eigenvalues

引入并讨论了对称箭头矩阵完成问题:在事先给定的对称箭头矩阵中嵌入一行一列使之成为新的对称箭头矩阵,并且具有指定最小最大特征值.利用箭头矩阵特征多项式之间的递归关系,给出并证明了这个问题存在惟一解的充要条件,以及解的一般公式与计算方法.同时还给出了存在非负解及均匀箭柄解的充要条件.利用该问题解决了逆特征值问题:求一个对称箭头矩阵,使它的各阶顺序主子阵具有给定的最小最大特征值.并给出该逆特征值问题解的计算方法.数值计算表明,该算法更有效.

The matrix completion problem was introduced and discussed： to reconstruct a new arrowhead matrix with specified minimal and maximal eigenvalues by embedding a row and column in a given arrowhead matrix. By the recurrence relations of the sequence of characteristic polynomials, the necessary and sufficient conditions of existing the unique solution and the non- negative solution for this problem are proved. And the expressions of the general solution are derived. According to this problem, the following inverse eigenvalue problem was solved： to find an arrowhead matrix from the minimal and maximal eigenvalues of all its leading principal submatrices. Furthermore, corresponding numerical algorithms and some examples are given. The numerical examples show that these numerical algorithms are more efficient.