Norm estimations for the Moore-Penrose inverse of the weak perturbation of matrices. Issue 2 (22nd January 2022)

• Record Type: Journal Article
• Authors: Fu, Chunhong; Song, Chuanning; Wang, Guorong; Xu, Qingxiang
• Is Part Of: Linear & multilinear algebra. Volume 70:Issue 2(2022)
• Page Start: 215
• Page End: 237
• Publication Date: 2022-01-22
• Subjects: Moore-Penrose inverse -- multiplicative perturbation -- weak perturbation -- strong perturbation -- acute perturbation; 15A09 -- 15A23 -- 15A60 -- 65F35; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2020.1717416
• Languages: English
• ISSNs: 0308-1087

Abstract

ABSTRACT: A multiplicative perturbation M of a matrix T has the form M = E T F ∗, where E and F are square matrices. It is proved that every acute perturbation is essentially a strong perturbation, which is a type of multiplicative perturbation. It is also proved that for every multiplicative perturbation M, M is a strong perturbation if and only if it is a weak perturbation and is rank-preserving. Some norm equations for the Moore-Penrose inverse are derived in the framework of the weak perturbation, through which some norm upper bounds for M † − T † are obtained. As an application, the perturbation estimation for the solution to the least squares problems is provided. The sharpness of the newly obtained upper bounds are illustrated by several numerical examples.