Acute perturbation bounds of weighted Moore–Penrose inverse. Issue 4 (3rd April 2018) - Ma, Haifeng
Abstract : It is well known that the upper bounds of the weighted Moore–Penrose inverse∥ A ¯ M, N † ∥ N, M ≤ ∥ A M, N † ∥ N, M 1 − ∥ A M, N † ∥ N, M ⋅ ∥ Δ A ∥ M, N, A ¯ = A + Δ A play a fundamental role in the perturbation analysis for the weighted linear least squares problem. In this note, we provide a sharp estimation for∥ A ¯ M, N † ∥ N, M, (1) ∥ A ¯ M, N † ∥ N, M ≤ ∥ ( I 2 r + Z Z T ) − 1 ∥ ∥ A M, N † ∥ N, M 1 − ∥ A M, N † ∥ N, M ∥ Δ A ∥ M, N, ∥ ( I 2 r + Z Z T ) − 1 ∥ < 1 if and only if R ( A ¯ ) ∩ R ( A ) = 0 and R ( A ¯ T ) ∩ R ( A T ) = 0 . Thus norm estimations for the weighted Moore–Penrose inverses of the acute perturbations can be improved uniformly.