On the square of a Jordan block

Let J be a Jordan block of size n and eigenvalue \lambda over a field K whose characteristic is not 2.

  1. Suppose \lambda \neq 0. Prove that the Jordan canonical form of J^2 is the Jordan block of size n with eigenvalue \lambda^2.
  2. Suppose \lambda = 0. Prove that the Jordan canonical form of J^2 has two blocks (with eigenvalue 0) of size n/2, n/2 if n is even and of size (n+1)/2, (n-1)/2 if n is odd.

First suppose \lambda \neq 0.

Lemma: Let e_i denote the ith standard basis element (i.e. e_i = [\delta_{i,1}\ \delta_{i,2}\ \ldots\ \delta_{i,n}]^\mathsf{T}. If 1 \leq i < n-2, then (J^2-\lambda^2I)e_{i+2} = 2\lambda e_{i+1} + e_i, (J^2-\lambda^2I)e_{2} = 2\lambda e_1, and (J^2-\lambda^2I)e_1 = 0. Proof: We have J^2-\lambda^2I = (J+\lambda I)(J-\lambda I). Evidently, (J-\lambda I)e_{i+2} = e_{i+1}. Now J+\lambda I = [b_{j,k}], where b_{j,k} = 2\lambda if j = k and 1 if k = j+1 and 0 otherwise. So (J+\lambda I)e_{i+1} = [\sum_k b_{j,k} \delta_{k,i+1}] = [b_{j,i+1}] = 2\lambda e_{i+1} + e_i if i > 1, and similarly (J^2-\lambda^2 I)(e_2) = 2\lambda e_1. \square

In particular, note that (J^2 - \lambda^2 I)^{n-1} e_n = 2^{n-1}\lambda^{n-1} e_1 is nonzero (here we use the noncharacteristictwoness of K), while (J^2 - \lambda^2 I)^n = 0. So the minimal polynomial of J^2 is (x-\lambda^2)^n. Thus the Jordan canonical form of J^2 has a single Jordan block, of size n, with eigenvalue \lambda^2.

Now suppose \lambda = 0. Evidently, we have Je_{i+1} = e_i and Je_1 = 0. Now J^n = 0 and J^{n-1} \neq 0. If n is even, we have (J^2)^{n/2} = 0 and (J^2)^{n/2-1} = J^{n-2} \neq 0, so that the minimal polynomial of J^2 is x^{n/2}. If n is odd, we have (J^2)^{(n+1)/2} = 0 while (J^2)^{(n+1)/2-1} = J^{n-1} \neq 0, so the minimal polynomial of J^2 is x^{(n+1)/2}.

Next, we claim that \mathsf{ker}\ J^2 has dimension 2. To this end, note that J^2e_{i+2} = e_i is nonzero, while J^2e_2 = J^2e_1 = 0. By this previous exercise, J^2 has 2 Jordan blocks; thus the blocks of J^2 both have eigenvalue 0 and are of size n/2,n/2 if n is even, and (n+1)/2, (n-1)/2 if n is odd.

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