The DJL Conjecture for CP Matrices over Special Inclines

TL;DR

This paper investigates the DJL conjecture for completely positive matrices over special inclines, proving its validity in these contexts and extending related factorization theorems.

Contribution

It demonstrates the DJL conjecture holds for matrices over certain special inclines and extends Markham's theorems to these settings.

Findings

The DJL conjecture is true for matrices over specific inclines.

Established sufficient conditions for triangular factorizations over special inclines.

Extended Markham's theorems to matrices over special inclines.

Abstract

Drew, Johnson and Loewy conjectured that for $n \geq 4$, the CP-rank of every $n \times n$ completely positive real matrix is at most $\left[n^{2}/4\right]$. While this conjecture is false for completely positive real matrices, we show that this conjecture is true for $n \times n$ completely positive matrices over certain special types of inclines. In addition, we prove an incline version of Markham's theorems which gives sufficient conditions for completely positive matrices over special inclines to have triangular factorizations.