Almost all integer matrices have no integer eigenvalues

2007/12/18 by Martin, Greg, Wong, Erick B. · 1 citation
#15A18 #15A36 #15A52 (Primary) 11C20 #60C05 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0712.3060

Abstract

For a fixed n≥2, consider an n× n matrix M whose entries are random integers bounded by k in absolute value. In this paper, we examine the probability that M is singular (hence has eigenvalue 0), and the probability that M has at least one rational eigenvalue. We show that both of these probabilities tend to 0 as k increases. More precisely, we establish an upper bound of size k^-2+ε for the probability that M is singular, and size k^-1+ε for the probability that M has a rational eigenvalue. These results generalize earlier work by Kowalsky for the case n=2 and answer a question posed by Hetzel, Liew, and Morrison.

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