Similarity to Symmetric Matrices over Finite Fields
It has been known for some time that every polynomial with coefficients from a finite field is the minimum polynomial of a symmetric matrix with entries from the same field. What have remained unknown, however, are the possible sizes for the symmetric matrices with a specified minimum polynomial and, in particular, the least possible size. In this paper we answer these questions using only the prime factorization of the given polynomial. Closely related is the question of whether or not a given matrix over a finite field is similar to a symmetric matrix over that field. Although partial results on that question have been published before, this paper contains a complete characterization.
Brawley, J. V., & Teitloff, T. C. (1998). Similarity to symmetric matrices over finite fields. Finite Fields and their Applications, 4(3), 261-274. doi:10.1006/ffta.1998.0216