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A $p$-adic Perron–Frobenius theorem

Volume 174 / 2016

Robert Costa, Patrick Dynes, Clayton Petsche Acta Arithmetica 174 (2016), 175-188 MSC: Primary 15B48; Secondary 15B51, 11S99, 37P20. DOI: 10.4064/aa8285-4-2016 Published online: 13 June 2016


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.


  • Robert CostaDepartment of Mathematics
    Tufts University
    503 Boston Avenue
    Medford, MA 02155, U.S.A.
  • Patrick DynesDepartment of Mathematical Sciences
    Clemson University
    O-110 Martin Hall
    Clemson, SC 29634, U.S.A.
  • Clayton PetscheDepartment of Mathematics
    Oregon State University
    Corvallis, OR 97331, U.S.A.

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