Event Detail

Event Type: 
Seminar
Date/Time: 
Wednesday, August 12, 2015 - 15:15 to 15:45
Location: 
Batcheller 250

Speaker Info

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.