Upper Bounds on Eigenvalue Variation

Let A and à = D
₁
*AD
₂
be two n × n diagonalizable matrices with eigendecomposition A = XΛX
^-1
and A = X̃Λ̃X̃
^-1
, where D
₁
, D
₂
, X and X̃ are nonsingular, and Λ = diag(λ1,⋯, λ
_n
) and Λ̃ = diag(λ̃
_1L
⋯, λ̃
_n
). Li [1] proved that if λ
₁
≥ λ
₂
≥ ⋯ ≥ λ
_n
≥ 0 and λ
₁
≥ λ
₂
≥ ⋯ ≥ λ
_n
≥ 0, then max
_{1 ≤ j ≤ n}
|λ
_j
-λ̃
_j
/λ
_j
| ≤ ∥X
^-1
∥
₂
∥X̃∥
₂
∥D
₂
∥
₂
× ∥X̃
^-1
(D
₁
*-D
₂
^-1
)X∥
₂
, max
_{1≤ j ≤ n}
|λ
_j
-λ̃
_j
/λ̃
_j
| ≤ ∥X
^-1
∥
₂
∥X̃∥
₂
∥D
₁
^-*
∥
₂
× ∥X̃
^-1
(D
₁
*-D
₂
^-1
)X∥
₂
. In this note, we show that the bounds are valid under slightly more general conditions.