Acute perturbation bounds of weighted Moore-Penrose inverse

It is well known that the upper bounds of the weighted Moore-Penrose inverse parallel to(A) over bar (dagger)(M,N)parallel to N,M <= parallel to(A) over bar (dagger)(M,N)parallel to N,M/1 - parallel to(A) over bar (dagger)(M,N)parallel to N,M . parallel to Delta A parallel to M,N , (A) over bar = A + Delta 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 parallel to(A) over bar (dagger)(M,N)parallel to N,M parallel to(A) over bar (dagger)(M,N)parallel to N,M <= parallel to(I-2r + ZZ(T))(-1)parallel to parallel to(A) over bar (dagger)(M,N)parallel to N,M/1 - parallel to(A) over bar (dagger)(M,N)parallel to N,M parallel to Delta A parallel to M,N, (1) parallel to(I-2r + ZZ(T))(-1)parallel to < 1 if and only if R(<(A)over bar>) boolean AND R(A) = {0} and R((A) over bar (T)) boolean AND R(A(T)) = {0}. Thus norm estimations for the weighted Moore-Penrose inverses of the acute perturbations can be improved uniformly.