We prove that if an n×n matrix defined over 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 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa8285-4-2016,
author = {Robert Costa and Patrick Dynes and Clayton Petsche},
title = {A p-adic Perron-Frobenius theorem},
journal = {Acta Arithmetica},
volume = {172},
year = {2016},
pages = {175-188},
zbl = {06602751},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8285-4-2016}
}
Robert Costa; Patrick Dynes; Clayton Petsche. A p-adic Perron-Frobenius theorem. Acta Arithmetica, Tome 172 (2016) pp. 175-188. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8285-4-2016/