The optimal perturbation bounds of the Moore–Penrose inverse under the Frobenius norm

Perturbation bounds for the Moore-Penrose inverse of operators

“…P. A. Wedin [24] presented some perturbation bounds for the Moore-Penrose inverse of matrices under general unitarily invariant norm, the spectral norm and the Frobenius norm. L. Meng and B. Zheng [16] obtained the optimal perturbation bounds for the Moore-Penrose inverse of matrices under the Frobenius norm using singular value decomposition and these results extended the results from [24]. C. Deng and Y. Wei [6] considered the perturbation bound for the Moore-Penrose inverse of operators on Hilbert spaces while the perturbation bounds of linear operators on Banach spaces have been considered in [18,26].…”

The optimal perturbation bounds for the weighted Moore-Penrose inverse

“…We note that a multiplicative perturbation can also be viewed as an additive one. The example in Remark 4.1 of [3] and Example 3 in [5] have shown that the multiplicative bounds (4.11), (4.13) are better than the additive bounds (3.29) and (3.26), respectively.…”

“…Examples 1 and 2 in [5] show the optimality of the additive perturbation bounds in (3.15) and (3.26), respectively. The example in Remark 3.3 of [3] shows the approximate optimality of the perturbation bound in (3.14).…”

“…In Sections 3 and 4, we consider the additive and multiplicative perturbation of the weighted Moore-Penrose inverse. Some new bounds for additive and multiplicative perturbation under the norms · (M N ) , · Q(M N ) and · F (M N ) are presented, which extends the corresponding ones in [5] and [13]. In Section 5, we give some numerical examples to illustrate the optimality of our given bounds under the weighted Q-norm and F -norm, respectively.…”

Multiplicative perturbation bounds of the group inverse and oblique projection

“…But in general this technique may produce unideal perturbation bounds because it overlooks the nature of the multiplicative perturbation. Various multiplicative perturbation analysis have been done to many problems, such as the polar decomposition [4], the singular value decomposition [5], and the Moore-Penrose inverse [7] when A is multiplicatively perturbed. In this paper, we will study the multiplicative perturbation bounds to the group inverse and the related oblique projection under unitarily invariant norm.…”