Assume we have a non-negative block square matrix $B$ as follows
$B=\left[ \begin{array}{c|c|c} \mathbf{0}&\mathbf{B}_{12}&\mathbf{B}_{13}\\ \hline \mathbf{0}&\mathbf{B}_{22}& \mathbf{B}_{23}\\ \hline \mathbf{0}&\mathbf{B}_{32}&\mathbf{B}_{33} \end{array} \right]$.
Can we still apply the Perron-Frobenius theorem on this matrix? What would be the impact of zero column blocks on the maximum positive eigenvalues and the corresponding left and right eigenvectors?
Experimentally, for the cases that I have tested, I found that the maximum eigenvalue, $\lambda_1$, is positive. The entries of the maximum left eigenvector $W_1$ corresponding to the zero columns of $B$ are all zero and the others were positive. Moreover, all the elements of right eigenvector $U_1$ were positive. I am wondering if there is any general result about this example?