正定Hermitian矩阵乘积特征值的新估计

NEW ESTIMATION OF THE EIGENVALUES OF AB FOR A>0, B>0

本文绘出了当A,B是正定Hermitian矩阵时,乘积AB的特征值的上、下界估计。所得结果改进了文[3]中相应结论。其结果如下:定理,设A>0,B>0且μ_1≥μ_2≥…≥μ_n,v_1≥v_2≥…≥v_n,λ_1≥λ_2≥…λ_n是A,B和AB的特征值。

For positive definite Hermitian matrices A and B, we obtain an upper and a lower bound for the eigenvalues of AB in terms of those of A and B, which improve the results in [3]. The results read as follows: Theorem let A>0, B>0 and are the eigenvalues of A, B and AB respectively. Then