# Perturbation bounds for $g$-inverses with respect to the unitarily invariant norm

Document Type: Research Paper

Author

College of Mathematics and Statistics‎, ‎Northwest Normal University‎, ‎Lanzhou 730070‎, ‎P.R‎. ‎China.

Abstract

Let complex matrices $A$ and $B$ have the same sizes. Using the singular value decomposition, we characterize the $g$-inverse $B^{(1)}$ of $B$ such that the distance between a given $g$-inverse of $A$ and the set of all $g$-inverses of the matrix $B$ reaches minimum under the unitarily invariant norm. With this result, we derive additive and multiplicative perturbation bounds of the nearest perturbed $g$-inverse. These results generalize and improve the existing results published recently to some extent.

Keywords

Main Subjects

### References

A. Ben-Israel, On error bounds for generalized inverses, SIAM J. Numer. Anal. 3 (1966), no. 4, 585--592.

A. Ben-Israel and T.N.E. Greville, Generalized Inverses: Theory and Applications, Springer, 2nd edition, New York, 2003.

N. Castro-Gonzalez, J. Ceballos, F. Dopico and J. Molera, Accurate solution of structured least squares problems via rank-revealing decompositions, SIAM. J. Matrix Anal. Appl. 34 (2013), no. 3, 1112--1128.

N. Castro-Gonzalez, F. Dopico and J. Molera, Multiplicative perturbation theory of the Moore-Penrose inverse and the least squares problem, Linear Algebra Appl. 503 (2016) 1--25.

N. Hu, W. Luo, C. Song and Q. Xu, Norm estimations for the Moore-Penrose inverse of multiplicative perturbations of matrices, J. Math. Anal. Appl. 437 (2016) 498--512.

L. Cai, W. Xu and W. Li, Additive and multiplicative perturbation bounds for the Moore-Penrose inverse, Linear Algebra Appl. 434 (2011) 480--489.

X. Liu, W. Wang and Y. Wei, Continuity properties of the f1g-inverse and perturbation bounds for the Drazin inverse, Linear Algebra Appl. 429 (2008) 1026--1037.

L. Meng and B. Zheng, The optimal perturbation bounds of the Moore-Penrose inverse under the Frobenius norm, Linear Algebra Appl. 432 (2010) 956--963.

L. Meng and B. Zheng, New multiplicative perturbation bounds of the Moore-Penrose inverse, Linear Multilinear Algebra 63 (2015), no. 5, 1037--1048.

L. Meng, B. Zheng and P. Ma, Perturbation bounds of generalized inverses, Appl. Math. Comput. 296 (2017) 88--100.

L. Mirsky, Symmetric gauge functions and unitarily invariant norms, Quart. J. Math. 11 (1960), no. 1, 50--59.

G. W. Stewart, On the continuity of the generalized inverses, SIAM J. Appl. Math. 17 (1969), no. 1, 33--45.

G.W. Stewart, On the perturbation of pseudo-inverses, projections, and linear least squares problems, SIAM Rev. 19 (1977), no. 4, 634--662.

J. Sun, Matrix Perturbation Analysis, Science Press, 2nd edition, Beijing, 2001.

G. Wang, Y. Wei and S. Qiao, Generalized Inverses: Theory and Computations, Science Press, Beijing, 2004.

P.A. Wedin, Perturbation theory for pseudoinverse, BIT 13 (1973), no. 2, 217--232.

M. Wei and S. Ling, On the perturbation bounds of g-inverse and oblique projections, Linear Algebra Appl. 433 (2010) 1778--1792.

P. Zhang and H. Yang, A note on multiplicative perturbation bounds for the Moore Penrose inverse, Linear Multilinear Algebra 62 (2014), no. 6, 831--838.

### History

• Receive Date: 30 June 2017
• Revise Date: 10 March 2018
• Accept Date: 10 March 2018