Eigenvalues of AB when A and B are Commutative Hermitian Matrices
Li Bo Luo · Acta Scientiarum Naturalium Universitatis Sunyatseni · 1999
Let A and B be n by n commutative Hermitian matrices with eigenvalues μ 1≤…≤μ n and γ 1≤…≤γ l0≤…≤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} .