Acute perturbation bounds of weighted Moore–Penrose inverse

Abstract

It is well known that the upper bounds of the weighted Moore–Penrose inverse $\parallel {\overline{A}}_{M,N}^{†}{\parallel }_{N,M}\le \frac{\parallel A_{M,N}^{†}{\parallel }_{N,M}}{1-\parallel A_{M,N}^{†}{\parallel }_{N,M}\cdot \parallel \mathrm{\Delta }A{\parallel }_{M,N}},\overline{A}=A+\mathrm{\Delta }A$ play a fundamental role in the perturbation analysis for the weighted linear least squares problem. In this note, we provide a sharp estimation for $\parallel {\overline{A}}_{M,N}^{†}{\parallel }_{N,M}$, (1) $\begin{array}{rl}& \parallel {\overline{A}}_{M,N}^{†}{\parallel }_{N,M}\le \parallel (I_{2r}+Z{Z}^{\mathrm{T}}{)}^{-1}\parallel \frac{\parallel A_{M,N}^{†}{\parallel }_{N,M}}{1-\parallel A_{M,N}^{†}{\parallel }_{N,M}\parallel \mathrm{\Delta }A{\parallel }_{M,N}},\\ & \parallel (I_{2r}+Z{Z}^{\mathrm{T}}{)}^{-1}\parallel <1\text{if and only if }\mathcal{R}\left(\overline{A}\right)\cap \mathcal{R}\left(A\right)=\left\{0\right\}\mathrm{and}\\ & \mathcal{R}\left({\overline{A}}^{\mathrm{T}}\right)\cap \mathcal{R}\left(A^{\mathrm{T}}\right)=\left\{0\right\}.\end{array}$(1) Thus norm estimations for the weighted Moore–Penrose inverses of the acute perturbations can be improved uniformly.

Keywords:

• Acute perturbation
• stable perturbation
• weighted Moore–Penrose inverse

2010 AMS Subject Classifications:

• 15A09
• 65F20

Acknowledgments

The author would like to thank the editor and two referees for their detailed comments which greatly improve the presentation of the paper. Partially work was finished during the author visited at Nis University, the author thanks Professor Dragana S. Cvetković-Ilić for her great hospitality.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

Partially work was finished during the author visited at Nis University which supported by National Natural Science Foundation of China under grants [11401143] and China Scholarship Council (Young Backbone Teachers Project 201607167005). Overseas Returning Foundation of Hei Long Jiang Province (Grant No. LC201402)