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Acta Mathematica Vietnamica

, Volume 43, Issue 4, pp 629–639 | Cite as

Completely Positive Matrices: Real, Rational, and Integral

  • Abraham Berman
  • Naomi Shaked-Monderer
Article
  • 131 Downloads

Abstract

This paper was presented as an invited talk in the 6th International Conference on Matrix Analysis and Applications, Duy Tan University, Da Nang City, Vietnam, June 15–18, 2017. All the matrices in the paper are real. We survey known results and present some new problems. The paper has six parts.
  1. 1.

    What is complete positivity?

     
  2. 2.

    Why are completely positive matrices important?

     
  3. 3.

    How can we tell if a given matrix is completely positive?

     
  4. 4.

    Does every rational completely positive matrix have a rational cp-factorization?

     
  5. 5.

    cp-rank.

     
  6. 6.

    Which integral completely positive matrices have an integral cp-factorization?

     

Keywords

Completely positive matrices CP-rank Copositive optimization 

Mathematics Subject Classification (2010)

15B48 15A23 

Notes

Funding Information

This work was supported by grant no. 2219/15 by the ISF-NSFC joint scientific research program.

References

  1. 1.
    Ando, T.: Completely Positive Matrices. Lecture Notes. The University of Wisconsin, Madison (1991)Google Scholar
  2. 2.
    Barioli, F.: Chains of dog-ears for completely positive matrices. Linear Algebra Appl. 330(1-3), 49–66 (2001)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barioli, F., Berman, A.: The maximal cp-rank of rank k completely positive matrices. Linear Algebra Appl. 363, 17–33 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ben-Haim, A., Weintraub, A.: Unpublished Report. Undergraduate Research Project, Technion (2016)Google Scholar
  5. 5.
    Berman, A.: Complete positivity. Linear Algebra Appl. 107, 57–63 (1988)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Berman, A., Dür, M., Shaked-Monderer, N.: Unpublished report (2014)Google Scholar
  7. 7.
    Berman, A., Grone, R.: Completely positive bipartite matrices. Math. Proc. Cambridge Philos. Soc. 103(2), 269–276 (1988)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Berman, A., Hershkowitz, D.: Combinatorial results on completely positive matrices. Linear Algebra Appl. 95, 111–125 (1987)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Berman, A., King, C., Shorten, R.: A characterisation of common diagonal stability over cones. Linear Multilinear Algebra 60(10), 1117–1123 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Berman, A., Plemmons, R. J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia (1994)CrossRefGoogle Scholar
  11. 11.
    Berman, A., Rothblum, U. G.: A note on the computation of the CP-rank. Linear Algebra Appl. 419(1), 1–7 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Berman, A., Shaked-Monderer, N.: Remarks on completely positive matrices. Linear Multilinear Algebra 44(2), 149–163 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Berman, A., Shaked-Monderer, N.: Completely Positive Matrices. World Scientific Publishing Co., Inc., River Edge (2003)CrossRefGoogle Scholar
  14. 14.
    Berman, A., Xu, C.: {0, 1} completely positive matrices. Linear Algebra Appl. 399, 35–51 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Bomze, I. M.: Copositive optimization–recent developments and applications. European J. Oper. Res. 216(3), 509–520 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Bomze, I. M., Schachinger, W., Ulrich, R.: New lower bounds and asymptotics for the cp-rank. SIAM J. Matrix Anal. Appl. 36(1), 20–37 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Bose, S., Slud, E.: Maximin efficiency-robust tests and some extensions. J. Statist. Plann. Inference 46(1), 105–121 (1995)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Cottle, R. W., Habetler, G. J., Lemke, C. E.: On classes of copositive matrices. Linear Algebra Appl. 3, 295–310 (1970)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Dahl, G., Haufmann, T. A.: Zero-one completely positive matrices and the A(R,S) classes. Spec. Matrices 4, 296–304 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Dickinson, P. J. C., Dür, M.: Linear-time complete positivity detection and the decomposition of sparse matrices. SIAM J. Matrix Anal. Appl. 33(3), 701–720 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Drew, J. H., Johnson, C. R.: The no long odd cycle theorem for completely positive matrices. Random discrete structures. IMA Math. Appl. 76, 103–115 (1996)zbMATHGoogle Scholar
  22. 22.
    Drew, J. H., Johnson, C. R., Loewy, R.: Completely positive matrices associated with M-matrices. Linear Multilinear Algebra 37(4), 303–310 (1994)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Dutour Sikirić, M., Schürmann, A., Vallentin, F.: Rational factorizations of completely positive matrices. Linear Algebra Appl. 523, 46–51 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Dür, M.: Copositive programming—a survey. In: Recent advances in optimization and its applications in engineering, pp 3–29 (2010)CrossRefGoogle Scholar
  25. 25.
    Dür, M., Still, G.: Interior points of the completely positive cone. Electron. J. Linear Algebra 17, 48–53 (2008)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Gray, L. J., Wilson, D. J.: Nonnegative factorization of positive semidefinite nonnegative matrices. Linear Algebra Appl. 31, 119–127 (1980)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Hall, M.: Combinatorial Theory. John Wiley-Interscience, 2nd edn (1998)Google Scholar
  28. 28.
    Hall, M., Newman, M.: Copositivity and completely positive quadratic forms. Proc. Camb. Phil. Soc. 59, 329–339 (1963)CrossRefGoogle Scholar
  29. 29.
    Hannah, J., Laffey, T. J.: Nonnegative factorization of completely positive matrices. Linear Algebra Appl. 55, 1–9 (1983)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Kaykobad, M.: On nonnegative factorization of matrices. Linear Algebra Appl. 96, 27–33 (1987)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Kelly, G.: A test of the Markovian model of DNA evolution. Biometrics 50(3), 653–664 (1994)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Kogan, N., Berman, A.: Characterization of completely positive graphs. Discrete Math. 114(1-3), 297–304 (1993)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Loewy, R., Tam, B. -S.: CP-Rank of completely positive matrices of order 5. Linear Algebra Appl. 363, 161–176 (2003)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Markham, T. L.: Factorization of nonnegative matrices. Proc. Am. Math. Soc. 32, 45–47 (1972)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Mason, O., Shorten, R.: On linear copositive Lyapunov functions and the stability of switched positive linear systems. IEEE Trans. Automat. Control 52(7), 1346–1349 (2007)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Maxfield, J. E., Minc, H.: On the matrix equation X T X = A. Proc. Edinburgh Math. Soc. 13, 125–129 (1963)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Miller, D. A., Zucker, S. W.: Copositive-plus Lemke algorithm solves polymatrix games. Oper. Res. Lett. 10(5), 285–290 (1991)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Motzkin, T.: Copositive quadratic forms. National Bureau of Standards Report 1818, 11–12 (1952)Google Scholar
  39. 39.
    Natarajan, K., Teo, C. P., Zheng, Z.: Mixed 0–1 linear programs under objective uncertainty: a completely positive representation. Oper. Res. 59(3), 713–728 (2011)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Shaked-Monderer, N.: A note on upper bounds on the cp-rank. Linear Algebra Appl. 431(12), 2407–2413 (2009)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Shaked-Monderer, N., Berman, A., Bomze, I. M., Jarre, F., Schachinger, W.: New results on the cp-rank and related properties of co(mpletely) positive matrices. Linear Multilinear Algebra 63(2), 384–396 (2015)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Shaked-Monderer, N., Bomze, M., Jarre, F, Schachinger, W.: On the cp-rank and minimal cp factorizations of a completely positive matrix. SIAM J. Matrix Anal. Appl. 34(2), 355–368 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsTechnion-Israel Institute of TechnologyHaifaIsrael
  2. 2.The Max Stern Yezreel Valley CollegeYezreel ValleyIsrael

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