Orderings on semirings and completely positive matrices

Abstract

A real symmetric matrix is called a completely positive matrix if there exists a nonnegative real matrix such that . In this paper, we extend the notion of complete positivity for matrices over real numbers to matrices over semirings in general. We find necessary and sufficient conditions for matrices over certain semirings to be completely positive. We also find an upper bound on the CP-rank of completely positive matrices over certain special types of semirings.

Keywords:

• matrices over semirings
• completely positive matrices
• diagonally dominant matrices
• inclines
• ordered semirings

AMS Subject Classifications:

• 15B33
• 15B48
• 06A75
• 16Y60

Acknowledgements

The contents of this paper form part of the first author’s doctoral thesis. The first author would also like to acknowledge the Ontario Graduate Scholarship (OGS Scholarship) supported by the province of Ontario. The second author acknowledges the support of NSERC discovery (grant number 400550).

Notes

No potential conflict of interest was reported by the authors.