Minimal dimension of symmetric or skew-symmetric matrices of given minimal polynomial

Over a field k of characteristic not 2 the set of minimal polynomials of symmetric or skew-symmetric matrices (with respect to an involution of the first kind) is known. We give the smallest possible dimension of a symmetric or skew-symmetric matrix of given minimal polynomial depending on the type of the involution. Concerning the transpose, we give the smallest constant c such that any suitable polynomial f is the minimal polynomial of a symmetric (resp. skew-symmetric) matrix of dimension c deg f. The case of polynomials of degree 2 is completely solved.