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

Martin, Greg; Wong, Erick

doi:10.2140/ant.2008.2.979

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The number of 2$\times$2 integer matrices having a prescribed integer eigenvalue

Greg Martin and Erick Wong

Vol. 2 (2008), No. 8, 979–1000

DOI: 10.2140/ant.2008.2.979

Abstract

What is the probability that an integer matrix chosen at random has a particular integer as an eigenvalue, or an integer eigenvalue at all? For a random real matrix, what is the probability of there being a real eigenvalue in a particular interval? This paper solves these questions for $2 \times 2$ matrices, after specifying the probability distribution suitably.

Keywords

random matrix, eigenvalue, integer eigenvalue, integer matrix, distribution of eigenvalues

Mathematical Subject Classification 2000

Primary: 15A36, 15A52

Secondary: 11C20, 15A18, 60C05

Milestones

Received: 28 May 2008

Revised: 13 August 2008

Accepted: 16 October 2008

Published: 23 November 2008

Authors

Greg Martin
Department of Mathematics University of British Columbia Room 121, 1984 Mathematics Road Vancouver, BC V6T 1Z2 Canada
http://www.math.ubc.ca/~gerg
Erick Wong
Department of Mathematics University of British Columbia Room 121, 1984 Mathematics Road Vancouver, BC V6T 1Z2 Canada
http://www.math.ubc.ca/~erick

Vol. 2, No. 8, 2008