A simplex algorithm for rational cp-factorization

TL;DR

This paper introduces a simplex-like algorithm that computes rational cp-factorizations of matrices, demonstrating that all integral 2x2 completely positive matrices have integral cp-factorizations, thus advancing understanding of matrix factorizations.

Contribution

The paper presents a novel simplex algorithm for rational cp-factorization, providing a constructive method and proving integral factorizations exist for 2x2 matrices.

Findings

Algorithm successfully finds rational cp-factorizations

All integral 2x2 completely positive matrices have integral cp-factorizations

Provides a new approach to matrix factorization problems

Abstract

In this paper we provide an algorithm, similar to the simplex algorithm, which determines a rational cp-factorization of a given matrix, whenever the matrix allows such a factorization. This algorithm can be used to show that every integral completely positive $2 \times 2$ matrix has an integral cp-factorization.