Linearized systems and a generalized Cayley-Hamilton theorem

We study linearized systems of equations in characteristic p not equal 0 of the form Ax = x((q)) where A is a square matrix and q = p(m). We present algorithms for calculating their solutions and for determining the minimum distance of their solution spaces. In the case when A has entries in F-q, the finite field of q elements, we explore the relationships between the minimal and characteristic polynomials of A and the above mentioned features of the solution space. In order to extend and generalize these findings to the case when A has entries in an arbitrary field of characteristic p, we obtain generalizations of the characteristic polynomial of a matrix and the Cayley-Hamilton theorem to square linearized systems.