Find all possible Jordan Canonical forms of $A^2$

Question

Finding Jordan Canonical forms seems pretty straightforward mostly, but this one threw me off:

Let $A\in M_n(\Bbb{F}) $ be a matrix with a minimal polynomial $m_A(t)=(t-\lambda)^n$.
Find all the possible Jordan Canonical forms of $A^2$.

I've looked through all my notes and material and didn't find a single example of finding a Jordan Canonical form for a matrix by any power. I know how to find it well enough for $A$ alone, but wouldn't know how raising it by the power of 2 (or any power) would affect that.

Any tips? Thanks!

Answer 1

Do this:

• Find for $A$ the Jordan canonical form $J$ (or forms if there is more than one).

• Note that if $A= P^{-1}J P $ then $A^2= P^{-1}J^2 P$.

• Compute $J^2$.

While $J^2$ is not in normal form you will see what its normal form, and thus the one of $A^2$, is easily.