Let $F$ be a field of characteristic not 2. We consider the following problems : Let $f\in F[X]$ a monic polynomial, is there a $n\in \N^*$ and an involution of first kind on $M_n(F)$ having a symmetric (resp. skew-symmetric) element of minimal polynomial $f$ ? And, if a such $n$ exists, how small can it be ? We also consider the case of symmetric (resp. skew-symmetric) matrices for the transposition.
At the end of the fifties, Krakowski determined which polynomial can be the minimal polynomial of skew-symmetric matrices. Ten years later, Bender answered to the second problem in the symmetric case. Finally, in 1994, Bass, Estes and Guralnick regard symmetric matrices with minimal poynomial on rings, specially over rings of integers.
We have proved that we can find sweetable involution (different of the transposition) on the matrices of dimension equal to the degree of the given polynomial or to twice this degree. The results depend on the type of the involution.
Then, we consider symmetric or skew-symmetric matrices for the transposition. We determine the smallest constant $\mu _s(k)$ (respectively $\mu _a(k)$) such that a polynomial $f$ suitable is the minimal polynomial of a symmetric (respectively skew-symmetric) matrice of dimension $\mu _s(k)\mathrm {deg}f$ (respectively $\mu _a(k)\mathrm {deg}f$), the value of these constants depends on whether one can write -1 as sum of squares in $k$ or not, i.e. of the level of $k$.