Acta Mathematica Vietnamica

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

Completely Positive Matrices: Real, Rational, and Integral

  • Abraham BermanEmail author
  • Naomi Shaked-Monderer


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.


  6. 6.

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



Completely positive matrices CP-rank Copositive optimization 

Mathematics Subject Classification (2010)

15B48 15A23 


Funding Information

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


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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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