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Counting matrices
over a finite field with all eigenvalues in the field
Lisa Kaylor and David Offner
Vol. 7 (2014), No. 5, 627–645
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Abstract |
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Given a finite field and a positive integer , we give a procedure to count the matrices with entries in with all eigenvalues in the field. We give an exact value for any field for values of up to , and prove that for fixed , as the size of the field increases, the proportion of matrices with all eigenvalues in the field approaches . 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 is similar to a matrix in Jordan canonical form, and so we proceed by enumerating the number of 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. |
Keywords
eigenvalues, matrices, finite fields, Jordan form
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Mathematical Subject Classification 2010
Primary: 05A05, 15A18, 15B33
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Milestones
Received: 8 May 2013
Revised: 31 January 2014
Accepted: 25 February 2014
Published: 1 August 2014
Communicated by Kenneth S. Berenhaut
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Authors |
| Lisa Kaylor | |
| Department of Mathematics and
Computer Science Westminster College 319 South Market Street New Wilmington, PA 16142 United States |
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| David Offner | |
| Department of Mathematics and
Computer Science Westminster College 319 South Market Street New Wilmington, PA 16142 United States |
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