Upper Bounds on Eigenvalue Variation

Let A and à = D₁*AD₂ be two n × n diagonalizable matrices with eigendecomposition A = XΛX^-1 and A = X̃Λ̃X̃^-1, where D₁, D₂, X and X̃ are nonsingular, and Λ = diag(λ1,⋯, λ_n) and Λ̃ = diag(λ̃_1L ⋯, λ̃_n). Li [1] proved that if λ₁ ≥ λ₂ ≥ ⋯ ≥ λ_n ≥ 0 and λ₁ ≥ λ₂ ≥ ⋯ ≥ λ_n ≥ 0, then max_{1 ≤ j ≤ n}|λ_j-λ̃_j/λ_j| ≤ ∥X^-1∥₂∥X̃∥₂∥D₂∥₂ × ∥X̃^-1 (D₁*-D₂^-1)X∥₂, max_{1≤ j ≤ n}|λ_j-λ̃_j/λ̃_j| ≤ ∥X^-1∥₂∥X̃∥₂∥D₁^-*∥₂ × ∥X̃^-1(D₁*-D₂^-1)X∥₂. In this note, we show that the bounds are valid under slightly more general conditions.