Combined perturbation bounds : I. Eigensystems and singular value decompositions

In this paper we present some new combined perturbation bounds of eigenvalues and eigensubspaces for a Hermitian matrix H, particularly in an asymptotic sense, δ₁₂²|| sin Θ (U₁, Ũ₁)||_F² + Σ_i=1 ^τ (λ_i - λ̃_i)² ≤ || ΔHU₁||_F² + O (||ΔHU ₁||_F⁴), where λ_i denotes the eigenvalues of H and U₁ the eigensubspace corresponding to the eigenvalues λ_i, i = 1, 2, . . . , r. The bound for each factor of eigensystems is optimal due to the sine theorem and the Hoffman-Wielandt theorem. In addition, combined perturbation bounds for singular value decompositions and combined perturbation bounds in some, more general, measures are also obtained. © 2007 Society for Industrial and Applied Mathematics.