Let be a Jordan block of size and eigenvalue over a field whose characteristic is not 2.
- Suppose . Prove that the Jordan canonical form of is the Jordan block of size with eigenvalue .
- Suppose . Prove that the Jordan canonical form of has two blocks (with eigenvalue 0) of size if is even and of size if is odd.
First suppose .
Lemma: Let denote the th standard basis element (i.e. . If , then , , and . Proof: We have . Evidently, . Now , where if and if and 0 otherwise. So if , and similarly .
In particular, note that is nonzero (here we use the noncharacteristictwoness of ), while . So the minimal polynomial of is . Thus the Jordan canonical form of has a single Jordan block, of size , with eigenvalue .
Now suppose . Evidently, we have and . Now and . If is even, we have and , so that the minimal polynomial of is . If is odd, we have while , so the minimal polynomial of is .
Next, we claim that has dimension 2. To this end, note that is nonzero, while . By this previous exercise, has 2 Jordan blocks; thus the blocks of both have eigenvalue 0 and are of size if is even, and if is odd.