Almost All Integer Matrices Have No Integer Eigenvalues

In a recent issue of this MONTHLY, Hetzel, Liew, and Morrison [4] pose a rather natural question: what is the probability that a random n× n integer matrix is diagonalizable over the field of rational numbers? Since there is no uniform probability distribution on Z, we need to exercise some care in interpreting this question. Specifically, for an integer k ≥ 1, let Ik = {−k,−k + 1, . . . , k − 1, k} be the set of integers with absolute value at most k. Since Ik is finite, we are free to choose each entry of an n × n matrix M independently and uniformly at random from Ik, with each value having probability 1/(2k + 1). The probability that M has a given property, such as being diagonalizable over Q, is then a function of the parameter k; we consider how this function behaves as k → ∞. In particular, if the probability converges to some limiting value then it is natural to think of this limit as the “probability” that a random integer matrix has that property. We refer the reader to the article of Hetzel et al for an interesting discussion of some of the issues raised by this interpretation of probability as a limit of finitary probabilities (in doing so we lose countable additivity and hence the measure-theoretic foundation of modern probability theory after Kolmogorov). From a pragmatic viewpoint, this cost is outweighed by the fact that many beautiful number-theoretic results are most naturally phrased in the language of probability: for instance, the celebrated Erdős-Kac theorem [1] states that the number of prime factors of a positive integer n behaves (in an appropriate limiting sense) as a normally-distributed random variable with mean and variance both equal to log log n. (In this article we always mean the natural logarithm when we write log.) For any given integers n ≥ 2 and k ≥ 1, the set of random n× n matrices with entries in Ik is a finite probability space; it will be convenient to compute probabilities simply by counting matrices, so we introduce some notation for them. LetMn(k) denote the set of all n × n matrices whose entries are all in Ik; then we are choosing matrices uniformly fromMn(k), which has cardinality exactly (2k + 1)n2 . The probability that a random matrix inMn(k) satisfies a particular property is simply the number of matrices inMn(k) with that property divided by (2k + 1) 2 . For a given integer λ, let Mn(k) denote the set of all matrices in Mn(k) that have λ as an eigenvalue. Note that in particular,Mn(k) is the subset of singular matrices inMn(k). Likewise, we denote set of the matrices in Mn(k) having at least one integer eigenvalue by Mn(k) = ⋃ λ∈ZMn(k). The probability that a random matrix in Mn(k) has an integer eigenvalue is thus |Mn(k)|/(2k + 1) 2 . Our main result affirms and strengthens a conjecture made in [4]: for any n ≥ 2, the probability that a random n× n integer matrix has even a single integer eigenvalue is 0. We furthermore give a quantitative upper bound on the decay rate of the probability as k increases. It will be extremely convenient to use “Vinogradov’s notation” to express this decay rate: we write f(k) g(k) if