A $p$-adic Perron–Frobenius theorem
Volume 174 / 2016
Acta Arithmetica 174 (2016), 175-188
MSC: Primary 15B48; Secondary 15B51, 11S99, 37P20.
DOI: 10.4064/aa8285-4-2016
Published online: 13 June 2016
Abstract
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.
