配乘矩阵特征值反问题可解的充分条件

Sufficient Conditions for the Solvability of Multiplicative Inverse Eigenvalue Problem

设Ａ为ｎ阶半正定Ｈｅｒｍｉｔｅ矩阵。求非负实对角矩阵ｃ，使得矩阵ＣＡ具有预先指定的非负实特征值。本文给出几组使这一反问题有解的充分条件，当ｎ＝２时，给出的这些条件又都成为该反问题可解的必要条件。

Let n nonnegative real numbers λ_1,λ_2……,λ_n and A, a positive semidefiniteHermitian matrix A of order n,be given. We present sufficient conditions for the solv-ability of muItiplicative inverse eigenvalue problem((MH),for short); find a nonnega-tive diagonal matrix C such that the matrix CA possesses eigenvalues λ_1,λ_2,···,λ_n.Wepay more attention to the effects of smaller components of{λ_i}.In the case n=2, thesufficient conditions are also necessary for(MH)to have solutions.