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On Semimonotone Matrices, \(R_0\)-Matrices and Q-Matrices

Abstract

In 1979, Pang proved that within the class of semimonotone matrices, \(R_0\)-matrices are Q-matrices and conjectured that the converse is also true. Jeter and Pye gave a counterexample when \(n=5\) for the converse; namely, they gave a semimonotone matrix that is in Q but not in \(R_0\). In this paper, we prove this conjecture for semimonotone matrices of order \(n \le 3\) and provide a counterexample when \( n> 3\), showing the sharpness of the result. We also provide an application of this result.

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Acknowledgements

We profusely thank the referees for their detailed comments which helped us in shortening the proof of our main theorem and improved the presentation of the manuscript. We also thank the editor for many useful comments and Professor Seetharama Gowda for making several valuable suggestions on an earlier draft of the manuscript.

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Correspondence to Gomatam Ravindran.

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Communicated by Alexey F. Izmailov.

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Parthasarathy, T., Ravindran, G. & Kumar, S. On Semimonotone Matrices, \(R_0\)-Matrices and Q-Matrices. J Optim Theory Appl (2022). https://doi.org/10.1007/s10957-022-02066-3

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  • DOI: https://doi.org/10.1007/s10957-022-02066-3

Keywords

  • Q matrices
  • \(R_0\) matrices
  • Semimonotone matrices
  • Copositive matrices
  • Principal pivot transform
  • Completely mixed games

Mathematics Subject Classification

  • 90C33
  • 91A05