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2014 Counting matrices over a finite field with all eigenvalues in the field
Lisa Kaylor, David Offner
Involve 7(5): 627-645 (2014). DOI: 10.2140/involve.2014.7.627


Given a finite field F and a positive integer n, we give a procedure to count the n×n matrices with entries in F with all eigenvalues in the field. We give an exact value for any field for values of n up to 4, and prove that for fixed n, as the size of the field increases, the proportion of matrices with all eigenvalues in the field approaches 1n!. As a corollary, we show that for large fields almost all matrices with all eigenvalues in the field have all eigenvalues distinct. The proofs of these results rely on the fact that any matrix with all eigenvalues in F is similar to a matrix in Jordan canonical form, and so we proceed by enumerating the number of n×n Jordan forms, and counting how many matrices are similar to each one. A key step in the calculation is to characterize the matrices that commute with a given Jordan form and count how many of them are invertible.


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Lisa Kaylor. David Offner. "Counting matrices over a finite field with all eigenvalues in the field." Involve 7 (5) 627 - 645, 2014.


Received: 8 May 2013; Revised: 31 January 2014; Accepted: 25 February 2014; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1297.15012
MathSciNet: MR3245840
Digital Object Identifier: 10.2140/involve.2014.7.627

Primary: 05A05 , 15A18‎ , 15B33

Keywords: Eigenvalues , finite fields , Jordan form , matrices

Rights: Copyright © 2014 Mathematical Sciences Publishers


Vol.7 • No. 5 • 2014
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