Polynomial matrices, splitting subspaces and Krylov subspaces over finite fields

Let T be a linear operator on vector space V of dimension n over Fq. For any divisor m n, an m-dimensional subspace W is T-splitting ifV=W⊕TW⊕⋯⊕Td−1W, where d=n/m. σ(m,d;T) denote the number subspaces. Determining for arbitrary open problem that closely related to another important Krylov spaces. We discuss this connection and give explicit formulae in case invariant factors satisfy certain degree conditions. A with enumeration polynomial matrices also discussed.