Let $A$ be a real symmetric matrix with positive coefficients. How can we prove that:
There exists a positive eigenvector $v>0$ (all $v_i>0$) associated with the greates-absolute-value eigenvalue $\lambda^*$, i.e $\ $ $|\lambda^*|\ge |\lambda_i| \ \ \ \ \ \ \forall \lambda_i \in \sigma (A)$
This is a unique (up to positive multiples) positive eigenvector. In other words, there are no positive eigenvectors for $A$ other than $v$ and its positive multiples.