The number of 2x2 integer matrices having a prescribed integer eigenvalue

TL;DR

This paper investigates the probability that a randomly chosen 2x2 integer matrix has a specific integer eigenvalue, providing precise results in the context of integer matrices.

Contribution

It offers a detailed analysis of the likelihood that 2x2 integer matrices possess a given integer eigenvalue, addressing a fundamental question in random matrix theory.

Findings

Derived explicit probability formulas for integer eigenvalues in 2x2 matrices

Established conditions under which integer eigenvalues occur

Provided asymptotic behavior of probabilities as matrix entries grow

Abstract

Random matrices arise in many mathematical contexts, and it is natural to ask about the properties that such matrices satisfy. If we choose a matrix with integer entries at random, for example, what is the probability that it will have a particular integer as an eigenvalue, or an integer eigenvalue at all? If we choose a matrix with real entries at random, what is the probability that it will have a real eigenvalue in a particular interval? The purpose of this paper is to resolve these questions, once they are made suitably precise, in the setting of 2x2 matrices.