Event Type:

Seminar

Date/Time:

Wednesday, August 12, 2015 - 15:15 to 15:45

Location:

Batcheller 250

Guest Speaker:

Robert Costa and Patrick Dynes

Institution:

REU Students

Abstract:

We prove a result for square matrices over the p-adic numbers akin to the Perron-Frobenius Theorem for square matrices over the real numbers. In particular, we show that if a square n-by-n matrix A has all entries p-adically close to 1, but bounded away from 0, then this matrix will possess a unique maximal eigenvalue such that: (a) the maximal eigenvalue is a p-adic integer, (b) this eigenvalue has an algebraic multiplicity of one, and (c) there exists an eigenvector associated to this eigenvalue with all entries p-adically close to 1. Furthermore, we show that the forward orbit of any vector x under iteration by A, normalized by the maximal eigenvalue, converges to the projection of x onto the eigenspace of this eigenvalue.

Host: