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T and S are linear transformations of a finite dimensional real vector space V with dimension 4, such that they commute. The minimal polynomial of S is $(t-1)^2(t-2)^2$

How do I prove that the characteristic polynomial $P_T (t)$ of T splits completely (factors into unique linear factors) and has at most two distinct roots?

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  • $\begingroup$ By "splits completely" you mean $P_T(t) = (X-\alpha_1)\ldots (X-\alpha_n)$ with distinct $\alpha_i$ ? $\endgroup$ – Gabriel Romon Apr 12 '16 at 20:13
  • $\begingroup$ Yup, that is indeed what I meant. $\endgroup$ – George Apr 12 '16 at 20:23
  • $\begingroup$ No, you probably don't mean that the $\alpha_i$ are distinct, since ohterwise $S=T$ is a counter-example. $\endgroup$ – Captain Lama Apr 12 '16 at 20:24
  • $\begingroup$ In that case, $P_T$ has exactly $n$ distinct roots, and you need to prove it has at most $2$ distinct roots... weird $\endgroup$ – Gabriel Romon Apr 12 '16 at 20:24
  • $\begingroup$ Screenshot of the question, from a past exam, in case I'm misinterpreting something to a large degree: prntscr.com/arhidp $\endgroup$ – George Apr 12 '16 at 20:32
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You can try to show that $S$ is similar to $\begin{pmatrix} A & 0 \\ 0 & B\end{pmatrix}$ where $A$ is the companion matrix of $(t-1)^2$ and $B$ is the companion matrix of $(t-2)^2$.

Then try to see what matrices commute with that.

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