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

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

2008-08-14

arxiv.org/abs/0808.1922v1

Mathematics

Probability

18 pages, 2 figures

Scientific paper

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.

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