Given λ a real eigenvalue of a real n-by-by matrix A whose eigenvector has only non-negative components, I must show that λ is bounded by the minimum and maximum sums of the columns of A, that is
$$\min_{1 \le k \le n} \sum\limits_{i=1}^n a_{ik} \le \lambda \le \max_{1 \le k \le n} \sum\limits_{i=1}^n a_{ik}$$
I don't know how to approach this problem.
It looks closely related to the Perron-Frobenius theorem, which asserts that for a positive matrix B, there is a eigenvector v with only positive components and an eigenvalue that satisfies the above inequation. Any ideas?