The DJL Conjecture for CP Matrices over Special Inclines

Drew, Johnson and Loewy conjectured that for $n\ge 4$, the CP-rank of every $n\times n$ completely positive real matrix is at most $\left[n^2\mathrm{/}4\right]$. 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.