Let be an matrix over with 1 on the first superdiagonal and 0 elsewhere. Compute . Next, suppose is an Jordan block with eigenvalue , and compute .
First, given a parameter , we define a matrix as follows: , where if and otherwise. (That is, is the matrix with 1 on the th superdiagonal and 0 elsewhere, and .)
Lemma: for . Proof: We proceed by induction. Certainly the result holds for and . Now suppose it holds for some . We have . Consider entry : . If , then . So this sum is . Now if , then the whole sum is 0; otherwise, it is 1. So in fact .
Now . As we saw in this previous exercise, powers of past the th are all zero; so in fact we have . Note that the indices of nonzero entries of the are mutually exclusive. So is the matrix whose entry is if and 0 otherwise.
Now let be the Jordan block with eigenvalue . Note that ; using this previous exercise, we have . (Where is as above.)