Given the set of square matricesM⊂Mn+m(C) that keep the subspace W = Cnx{0} ⊂ Cn+m invariant, we obtain the implicit form of a miniversal deformation of a matrix a∈M, and we compute it explicitely when this matrix is marked (this is, if there is a permutation matrix p ∈ Mn+m(C) such that p−1ap is a Jordan matrix). We derive some applications to tackle the classical Carlson problem.