Positivity

, Volume 22, Issue 1, pp 379–398 | Cite as

Semipositive matrices and their semipositive cones

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Abstract

The semipositive cone of \(A\in \mathbb {R}^{m\times n}, K_A = \{x\ge 0\,:\, Ax\ge 0\}\), is considered mainly under the assumption that for some \(x\in K_A, Ax>0\), namely, that A is a semipositive matrix. The duality of \(K_A\) is studied and it is shown that \(K_A\) is a proper polyhedral cone. The relation among semipositivity cones of two matrices is examined via generalized inverse positivity. Perturbations and intervals of semipositive matrices are discussed. Connections with certain matrix classes pertinent to linear complementarity theory are also studied.

Keywords

Semipositive matrix Proper cone Polyhedral cone Generalized inverse positivity Linear complementarity Singular M-matrix Q-matrix 

Mathematics Subject Classification

15A48 90C33 15A23 15A09 

Notes

Acknowledgements

The first author thanks the Office of Alumni and International Relations, IIT Madras and the Department of Mathematics and Statistics, WSU for partial financial support for his visit to WSU in the winter of 2015. The authors thank M.S. Gowda of UMBC for his comments and suggestions, especially concerning Theorem 3.3. They also thank T. Parthasarathy for discussions on the linear complementarity problem.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia
  2. 2.Department of Mathematics and StatisticsWashington State UniversityPullmanUSA

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