  We prove that if an $n\times n$ matrix defined over ${\mathbb Q}_p$ (or more
generally an arbitrary complete, discretely-valued, non-Archimedean field)
satisfies a certain congruence property, then it has a strictly maximal
eigenvalue in ${\mathbb Q}_p$, and that iteration of the (normalized) matrix
converges to a projection operator onto the corresponding eigenspace. This
result may be viewed as a $p$-adic analogue of the Perron-Frobenius theorem for
positive real matrices.
