On the DJL conjecture for order 6

TL;DR

This paper advances the understanding of the DJL conjecture for 6x6 matrices by establishing an upper bound on the cp-rank for matrices orthogonal to certain extremal copositive matrices.

Contribution

It proves that 6x6 completely positive matrices orthogonal to exceptional extremal copositive matrices have cp-rank at most 9, moving closer to resolving the conjecture for n=6.

Findings

If A is a 6x6 completely positive matrix orthogonal to an exceptional extremal copositive matrix, then cp-rank(A) ≤ 9.

The result narrows the gap in the DJL conjecture for n=6.

Progress towards proving the conjecture for the critical case n=6.

Abstract

In 1994 Drew, Johnson and Loewy conjectured that for $n \ge 4$, the cp-rank of any $n\times n$ completely positive matrices is at most $\lfloor{n^2}/{4}\rfloor$. Recently this conjecture has been proved for $n=5$ and disproved for $n\ge 7$, leaving the case $n=6$ open. We make a step toward proving the conjecture for $n=6$. We show that if $A$ is a $6\times 6$ completely positive matrix that is orthogonal to an exceptional extremal copositive matrix, then the cp-rank of $A$ is at most $9$.