First of all, thank you! The approach you suggested looks very interesting and thought-provoking; maybe it can even be extended to the case of many matrices $A_1\dots A_m$. Now, about the details to be filled in: the first one I clear, it is not difficult to do. But I am a little puzzled by the second one, and I would greatly appreciate if you could elucidate it a little. What do you mean by an "analogue of the companion matrix"? Companion matrix to which polynomial? Did you mean just the convenient form that the companion matrix has, or do I miss something in your answer?