New multiplicative perturbation bounds of the Moore–Penrose inverse
Lingsheng Meng, Bing Zheng · Linear and Multilinear Algebra · 2014
Based on the singular value decomposition, we obtain new multiplicative perturbation bounds of the Moore–Penrose inverse under the Frobenius norm, the unitarily invariant norm and the -norm, respectively. These bounds always improve the existing bounds. An example is given to show that the bound under the Frobenius norm is optimal. Moreover, the optimal multiplicative perturbation bounds under the unitarily invariant norm and the -norm are, respectively, given out when the matrix has full column rank. In addition, we prove that Yang and Zhang [A note on multiplicative perturbation bounds for the Moore–Penrose inverse, Linear and Multilinear Algebra, 2014; 62:831–838] cannot show their bounds are optimal by using their examples.