Almost all integer matrices have no integer eigenvalues
| dc.creator | Martin, Greg | |
| dc.creator | Wong, Erick B. | |
| dc.date | 2007-12-18 | |
| dc.date.accessioned | 2026-07-07T08:50:15Z | |
| dc.date.available | 2026-07-07T08:50:15Z | |
| dc.description | For a fixed $n\ge2$, consider an $n\times n$ 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. | |
| dc.description | 9 pages, 1 figure | |
| dc.identifier | https://arxiv.org/abs/0712.3060 | |
| dc.identifier | http://arxiv.org/abs/0712.3060 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/144544 | |
| dc.subject | Number Theory | |
| dc.subject | 15A36, 15A52 (Primary) 11C20, 15A18, 60C05 (Secondary) | |
| dc.title | Almost all integer matrices have no integer eigenvalues | |
| dc.type | text |