Vol. 2, No. 8, 2008

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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
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 × 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