Similarity to symmetric matrices over finite fields
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.
scientific article; zbMATH DE number 1236200