可交换厄米特矩阵乘积的特征值

Eigenvalues of AB when A and B are Commutative Hermitian Matrices

设Ａ，Ｂ为ｎ阶不定厄米特矩阵，且ＡＢ＝ＢＡ；μｉ，γｉ及λｉ分别为Ａ，Ｂ及ＡＢ依升序排列的特征值．给出的上界λｋ≤（μｌ－ｋ＋１－μ１）γｌ＋μ１γ１（ｋ＝１，…，ｌ）及下界λ≥（μｋ－ｌ－μ１）γｌ＋１＋μ１γｎ（ｋ＝ｌ＋１，…，ｎ）（其中ｌ是Ｂ的负惯性指标）以及一系列结果改进了一般估计：ｍｉｎ｛μ１γｎ，μｎγ１｝≤λｋ≤ｍａｘ｛μ１γ１，μｎγｎ｝．

Let A and B be n by n commutative Hermitian matrices with eigenvalues μ 1≤…≤μ n and γ 1≤…≤γ l<0≤…≤γ n , respectively. Denote by λ 1≤…≤λ n the eigenvalues of AB. A series of bounds for λ k including (1) λ k≤(μ l-k+1 -μ 1)λ l+μ 1γ 1 (k=1,…,l) and (2) λ k≥(μ k-l -μ 1)γ l+1 +μ 1γ n (k=l+1,…,n) , are derived. These results are tighter than those in literature: min {μ 1γ n,μ nγ 1}≤λ k≤ max {μ 1γ 1,μ nγ n} .