Let be a field of characteristic 2. Compute the Jordan canonical form of a Jordan block of size and eigenvalue over . Characterize those matrices over which are squares; that is, characterize such that for some matrix .
Let be the Jordan block with eigenvalue and size . That is, if , if , and 0 otherwise. Now ; if or , then . Evidently then we have if , if , and 0 otherwise, Noting that . So , where is the zero matrix and is the identity matrix. Now let , where has dimension . Now . That is, ‘shifts’ the entries of – so and . In particular, the kernel of has dimension 2, so that by this previous exercise, the Jordan canonical form of has two blocks (both with eigenvalue .
Now , since has characteristic 2. Note that has order , since (evidently) we have and . So has order . If is even, then while , and if is odd, then while . So the minimal polynomial of is if is even and if is odd.
So the Jordan canonical form of has two Jordan blocks with eigenvalue . If is even, these have size , and if is odd, they have size .
Now let be an arbitrary matrix over (with eigenvalues in ). We claim that is a square if and only if the following hold.
- The eigenvalues of are square in
- For each eigenvalue of , the Jordan blocks with eigenvalue can be paired up so that the sizes of the blocks in each pair differ by 0 or 1.
To see the ‘if’ part, suppose is in Jordan canonical form, where and are Jordan blocks having the same eigenvalue and whose sizes differ by 0 or 1. By the first half of this exercise, is the Jordan canonical form of , where is a Jordan block with eigenvalue . Now is similar to the direct sum of these , and so . Then is square.
Conversely, suppose is square, and say is in Jordan canonical form. So . Letting denote the Jordan blocks of , we have . The Jordan canonical form of has two blocks with eigenvalue and whose sizes differ by 0 or 1, by the first half of this exercise. So the Jordan blocks of all have eigenvalues which are square in and can be paired so that the sizes in each pair differ by 0 or 1.