Similarity to symmetric matrices over finite fields
type: Item
instance of: scholarly article
title: Similarity to symmetric matrices over finite fields
zbMATH Open document ID: 0923.15009
author: Timothy C. Teitloff
author: Joel V. Brawley
published in: Finite Fields and their Applications
publication date: 1999-10-31T00:00:00Z
review text: The authors solve the following problems. (1) Let \(f\) be a polynomial \(f\) over \(F_q\), the finite field with \(q\) elements. Determine whether or not \(f\) is the minimum polynomial of a symmetric matrix, and, if it is, find the possible sizes of such matrices. (2) Characterize those matrices over \(F_q\) that are similar to symmetric matrices over \(F_q\). For (1), the answer is that a monic polynomial \(f\) of degree \(n\) is the minimum polynomial of a symmetric matrix if and only if one of the following holds: \(q \not\equiv 3\bmod 4\); \(n\not\equiv 4\bmod 4\); \(f\) is not a square. The smallest size of such a symmetric matrix is then \(n+r\) where \(r\) is the smallest degree of an irreducible factor of \(f\). For (2), an \(n\times n\) matrix \(B\) over a \(F_q\) is similar to an \(n\times n\) symmetric matrix if and only if one of the following holds: \(q\not\equiv 3\bmod 4\); \(n\not\equiv 2\bmod 4\); \(B\) has an elementary divisor of degree not congruent to \(2\bmod 4\); \(B\) has an elementary divisor that is not a square.
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 11T06
zbMATH DE Number: 1236200
zbMATH Keywords: minimum polynomial
zbMATH Keywords: symmetric matrix
zbMATH Keywords: similarity of matrices
zbMATH Keywords: finite field
zbMATH Keywords: elementary divisor
MaRDI profile type: Publication
full work available at URL: ffta.1998.0216
OpenAlex ID: W2025623739
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DOI: 10.1006/FFTA.1998.0216
description: scientific article; zbMATH DE number 1236200