For strictly lower and upper triangular matrices $L$ and $U$, let $B=L+U$ be a nonnegative and irreducible $n\times n$ matrix. Prove that the spectral radius of matrix $H=(I-L)^{-1}U$ is positive and that its associated eigenvector has positive components?
Since the matrix $L$ is strictly lower triangular matrix, it can be proved that the matrix $H$ is also nonnegative matrix. They there exists a nonnegative vector $x\geq 0$ such that $Hx=\lambda x$, where $\lambda=\rho(H)\geq 0$. But the problem is how to use the nonnegative and irreducible character of matrix $B$ to show $\lambda$ and $x$ are both positive？