Valentin Dannenberg ; Achill Schürmann - Perfect Copositive Matrices

cm:11141 - Communications in Mathematics, July 16, 2023, Volume 31 (2023), Issue 2 (Special issue: Euclidean lattices: theory and applications) - https://doi.org/10.46298/cm.11141
Perfect Copositive MatricesArticle

Authors: Valentin Dannenberg ; Achill Schürmann

    In this paper we give a first study of perfect copositive $n \times n$ matrices. They can be used to find rational certificates for completely positive matrices. We describe similarities and differences to classical perfect, positive definite matrices. Most of the differences occur only for $n \geq 3$, where we find for instance lower rank and indefinite perfect matrices. Nevertheless, we find for all $n$ that for every classical perfect matrix there is an arithmetically equivalent one which is also perfect copositive. Furthermore we study the neighborhood graph and polyhedral structure of perfect copositive matrices. As an application we obtain a new characterization of the cone of completely positive matrices: It is equal to the set of nonnegative matrices having a nonnegative inner product with all perfect copositive matrices.


    Volume: Volume 31 (2023), Issue 2 (Special issue: Euclidean lattices: theory and applications)
    Published on: July 16, 2023
    Accepted on: June 29, 2023
    Submitted on: March 31, 2023
    Keywords: Mathematics - Metric Geometry,Mathematics - Number Theory,Mathematics - Optimization and Control,11H55 (Primary) 11H50, 90C20 (Secondary)

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