Minus partial order on regular matrices

Abstract

The theory of ‘minus partial order’ on the class of matrices over a field is well studied in the literature, and it is known that the rank additive property ‘’ holds whenever is lesser than under the minus partial order. The rank additive property fails in the class of regular matrices over a commutative ring, though several other characterizations of minus partial order relation known for the class of matrices over a field are easily extended. So, an extension of rank additive property in the class of regular matrices is further investigated. In the process, Rao–Mitra’s theorem on invariance of is further probed and a general condition for such invariance is obtained for matrices over a commutative ring.

Keywords:

• regular matrix
• generalized inverse
• minus partial order
• rank additivity

AMS Subject Classifications:

• 15A09
• 06A06

Acknowledgements

Authors are grateful to anonymous Referee and the editor J.F. Queiró for their comments and suggestions which helped in improving the contents at several places.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research work of first author was supported by ‘Science and Engineering Research Board (DST, Govt. of India)’ under Extra Mural Research Funding Scheme (SR/S4/MS:870/14).