Generating systems of the full matrix algebra that contain nonderogatory matrices

Let $\mathcal{A}$ be an algebra over a field $\mathbb{F}$ generated by a set of matrices $\mathcal{S}$. The paper considers algorithmic aspects of checking whether $\mathcal{A}$ coincides with the full matrix algebra. Laffey has shown that for $\mathbb{F} = \mathbb{C}$, under the assumption that $\mathcal{S}$ contains a Jordan matrix from a certain class, there is a fast method for checking whether $\mathcal{A}$ possesses nontrivial invariant subspaces and, consequently, coincides with the full algebra by Burnside's theorem. This paper extends the class to the largest subclass of Jordan matrices on which the algorithm works correctly. Examples demonstrating the different behavior of other matrix systems are provided.