不可约非负矩阵的逆特征值问题

非负矩阵逆特征值问题的提法是:对已知的一个复数组Λ={λ1,…,λn},求一个n×n非负矩阵以Λ为谱.由于非负矩阵逆特征值问题的理论兴趣和应用背景,长期以来,一直吸引不少研究者从事这个热门课题.论文对n=3的情形,限制在至少有三个零元的不可约矩阵类中.首先,给出具有已知的对角元集的非负矩阵逆特征值(包含复特征值)问题有解的充分必要条件;其次,在此基础上,更进一步证明非负矩阵逆特征值问题有解的充分必要条件.在两种情形下都给出了构造全部解集合的简单而有效的公式.

On inverse eigenvalue problem of irreducible nonnegative matrices

The nonnegative inverse eigenvalue problem is the problem of finding a nonnegative matrix with a given set Λ of complex numbers as its spectra.Due to its theoretical interest and applicative background,the nonnegative inverse eigenvalue problem always attracts a lot of researchers to work on it.Here we first proved the sufficient and necessary conditions for an irreducible nonnegative 3×3 matrix with at least three zero entries to have the given spectra(including complex eigenvalues) and the given set of diagonal entries.Then we proved the sufficient and necessary conditions for an irreducible nonnegative 3×3 matrix with at least three zero entries to have the given set Λ as its spectra.In both cases,simple formulas of the solution nonnegative matrices were given whenever they exist.