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BY 4.0 license Open Access Published by De Gruyter Open Access December 17, 2019

The integer cp-rank of 2 × 2 matrices

  • Thomas Laffey and Helena Šmigoc EMAIL logo
From the journal Special Matrices

Abstract

We show the cp-rank of an integer doubly nonnegative 2 × 2 matrix does not exceed 11.

MSC 2010: 15B36; 15B48

References

[1] Abraham Berman. Completely positive matrices – real, rational and integral. Mathematisches Forschungsinstitut Oberwolfach Report No. 52/2017, Copositivity and Complete Positivity, 2017.Search in Google Scholar

[2] Abraham Berman, Mirjam Dür, and Naomi Shaked-Monderer. Open problems in the theory of completely positive and copositive matrices. Electron. J. Linear Algebra, 29:46–58, 2015.10.13001/1081-3810.2943Search in Google Scholar

[3] Abraham Berman and Naomi Shaked-Monderer. Completely positive matrices. World Scientific Publishing Co., Inc., River Edge, NJ, 2003.10.1142/5273Search in Google Scholar

[4] Immanuel M. Bomze. Copositive optimization—recent developments and applications. European J. Oper. Res., 216(3):509–520, 2012.10.1016/j.ejor.2011.04.026Search in Google Scholar

[5] Immanuel M. Bomze, Werner Schachinger, and Gabriele Uchida. Think co(mpletely)positive! Matrix properties, examples and a clustered bibliography on copositive optimization. J. Global Optim., 52(3):423–445, 2012.10.1007/s10898-011-9749-3Search in Google Scholar

[6] Mathieu Dutour Sikiri¢, Achill Schürmann, and Frank Vallentin. A simplex algorithm for rational cp-factorization, 2018. https://arxiv.org/abs/1807.01382Search in Google Scholar

[7] John H. Drew and Charles R. Johnson. The no long odd cycle theorem for completely positive matrices. In Random discrete structures (Minneapolis, MN, 1993), volume 76 of IMA Vol. Math. Appl., pages 103–115. Springer, New York, 1996.10.1007/978-1-4612-0719-1_7Search in Google Scholar

[8] John H. Drew and Charles R. Johnson. The completely positive and doubly nonnegative completion problems. Linear and Multilinear Algebra, 44(1):85–92, 1998.10.1080/03081089808818550Search in Google Scholar

[9] John H. Drew, Charles R. Johnson, Steven J. Kilner, and Angela M. McKay. The cycle completable graphs for the completely positive and doubly nonnegative completion problems. Linear Algebra Appl., 313(1-3):141–154, 2000.10.1016/S0024-3795(00)00110-5Search in Google Scholar

[10] John H. Drew, Charles R. Johnson, and Fumei Lam. Complete positivity of matrices of special form. Linear Algebra Appl., 327(1-3):121–130, 2001.10.1016/S0024-3795(00)00315-3Search in Google Scholar

[11] Mathieu Dutour Sikirić, Achill Schürmann, and Frank Vallentin. Rational factorizations of completely positive matrices. Linear Algebra Appl., 523:46–51, 2017.10.1016/j.laa.2017.02.017Search in Google Scholar

[12] Marshall Hall, Jr. A survey of combinatorial analysis. In Some aspects of analysis and probability, Surveys in Applied Mathematics. Vol. 4, pages 35–104. John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958.Search in Google Scholar

[13] Thomas J. Laffey and Helena Šmigoc. Integer completely positive matrices of order two. Pure Appl. Funct. Anal., 3(4):633–638, 2018.Search in Google Scholar

[14] John E. Maxfield and Henryk Minc. On the matrix equation XX = A. Proc. Edinburgh Math. Soc. (2), 13:125–129, 1962/1963.10.1017/S0013091500014681Search in Google Scholar

Received: 2019-09-27
Accepted: 2019-12-01
Published Online: 2019-12-17

© 2019 Thomas Laffey et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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