Singular linear preservers of majorization and cone type majorization

Abstract

Since the introduction of majorization in ${\mathbb{R}}^n$ by Hardy, Littlewood and Polya, several extensions of this concept have been studied in the literature. Recently, Kosuru and Saha [Cone type majorization and its strong linear preservers. Electron J Linear Algebra. 2020;36(36):511–518.] defined the concept of cone type majorization. In this paper, we focus on the study of the behavior of the linear preservers of majorization and cone type majorization under generalized inversion, namely, Drazin inversion and Moore-Penrose inversion. A characterization of these linear preservers, given by Ando [Majorization, doubly stochastic matrices and comparisons of eigenvalues. Linear Algebra Appl. 1989;118:163–248.] for majorization, and by Kosuru and Saha [Cone type majorization and its strong linear preservers. Electron J Linear Algebra. 2020;36(36):511–518.] for cone type majorization, prove to be crucial in our proofs.

Keywords:

• Majorization
• cone type majorization
• linear preserver
• Drazin inverse
• Moore-Penrose inverse

2010 MSC:

• 15A09
• 15A86
• 15A23

Acknowledgements

We would like to thank the anonymous referee for the helpful comments that contributed to the improvement of the first version of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author acknowledges receiving partial funding from “Proyecto de I+D+i PID2019-106362GB-I00 financiado por MCIN/AEI/10.13039/501100011033”. The work of the second author was partially supported by “FCT-Fundação para a Ciência e Tecnologia, under project UIDB/04721/2020”, while the third author acknowledges funding from “MATRICS of SERB (MTR/2018/001132), Government of India”.