Expression for the multiplicative perturbation of the Moore–Penrose inverse

Abstract

In this paper, we study the multiplicative perturbation of an operator of the form , where X and Y are invertible. Based on this result, we study the Moore–Penrose invertibility of block matrix M with Schur complement having closed range. We recover the cases when a Schur complement in M is either equal to zero or nonsingular. The necessary and sufficient conditions for the existence as well as the expressions for the Moore–Penrose inverses are obtained.

Keywords:

• Block matrix
• Moore–Penrose inverse
• Schur complement

AMS Subject Classifications:

• 15A09
• 47A05

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

Supported by the Innovation Project of Graduate School of South China Normal University [grant number 2016YN08]; the National Natural Science Foundations of China [grant numbers 11671261, 11501345].