Almost all integer matrices have no integer eigenvalues

Details Almost all integer matrices have no integer eigenvalues Almost all integer matrices have no integer eigenvalues

2007-12-18

arxiv.org/abs/0712.3060v1

Mathematics

Number Theory

9 pages, 1 figure

Scientific paper

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 $k^{-2+\epsilon}$ for the probability that $M$ is singular, and size $k^{-1+\epsilon}$ 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.

Affiliated with

Also associated with

No associations

Rating

Almost all integer matrices have no integer eigenvalues does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Almost all integer matrices have no integer eigenvalues, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Almost all integer matrices have no integer eigenvalues will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-145846