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ELA

AN IMPROVED CHARACTERISATION OF THE INTERIOR OF

THE COMPLETELY POSITIVE CONE

∗

PETER J.C. DICKINSON†

Abstract. A symmetric matrix is defined to be completely positive if it allows a

factorisa-tionBBT, whereBis an entrywise nonnegative matrix. This set is useful in certain optimisation

problems. The interior of the completely positive cone has previously been characterised by D¨ur and Still [M. D¨ur and G. Still, Interior points of the completely positive cone,Electronic Journal of Linear Algebra, 17:48–53, 2008]. In this paper, we introduce the concept of the set of zeros in the nonnegative orthant for a quadratic form, and use the properties of this set to give a more relaxed characterisation of the interior of the completely positive cone.

Key words. Completely positive matrices, Copositive matrices, Cones of matrices.

AMS subject classifications. 15A23, 15B48.

∗Received by the editors on March 3, 2010. Accepted for publication on October 16, 2010.

Handling Editor: Moshe Goldberg.

†Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen,

P.O. Box 407, 9700 AK Groningen, The Netherlands ([email protected]). Electronic Journal of Linear Algebra ISSN 1081-3810

A publication of the International Linear Algebra Society Volume 20, pp. 723-729, November 2010

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