A matrix $A \in \mathbb{R}^{n \times n}$ is called completely positive if there exists an entrywise nonnegative matrix $B \in \mathbb{R}^{n \times r}$ such that $A = BB^{T}$. All eigenvalues of $A$ are real and nonnegative.

My question is when will a completely positive matrix have all positive eigenvalues?

The only completely positive matrix I know so far have zero eigenvalues is \begin{equation} A = \begin{pmatrix} 41 & 43 & 80 & 56 & 50 \\ 43 & 62 & 89 & 78 & 51 \\ 80 & 89 & 162 & 120 & 93 \\ 56 & 78 & 120 & 104 & 62 \\ 50 & 51 & 93 & 62 & 65 \end{pmatrix} . \end{equation} So probably that the case completely positive matrix has zero eigevalues are rare. But I could not find any documment on this.

**Update:** due to @Robert Israel answer.

Usually, we do not know $B$ in general and indeed the composition may not unique. Therefore the condition depends only in $A$ would be easier to verify. For example with \begin{equation} A = \begin{pmatrix} 18 & 9 & 9 \\ 9 & 18 & 9 \\ 9 & 9 & 18 \end{pmatrix} \end{equation} there are at least three decompose satisfies $B \geq 0$. In particular \begin{equation} B_{1} = \begin{pmatrix} 4 & 1 & 1 \\ 1 & 4 & 1 \\ 1 & 1 & 4 \end{pmatrix} , \quad B_{2} = \begin{pmatrix} 3 & 3 & 0 & 0 \\ 3 & 0 & 3 & 0 \\ 3 & 0 & 0 & 3 \end{pmatrix} , \quad B_{3} = \begin{pmatrix} 3 & 3 & 0 \\ 3 & 0 & 3 \\ 0 & 3 & 3 \end{pmatrix} . \end{equation}