This is in fact also the minimal polynomial and the characteristic polynomial : indeed divides which divides , and since the first and last are of degree and all are monic, they must all be the same. Another reason is that in general if any polynomial in annihilates a vector , then it also annihilates (just apply to the equation that says that it annihilates ), and therefore by iteration it annihilates the entire space generated by the iterated images by of ; in the current case we have seen that for that space is all of , so . Indeed one verifies for the full matrix that is the null matrix: