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Let $V$ be a $4$-dimensional real vector space and let $T:V \rightarrow V$ be an endomorphism such that its characteristic polynomial $\chi_T$ is given by $\chi_T = (t-2)^3t$. Suppose also that $V$ is $T$-cylic, that is to say, $V = \{p(T)(v_0): p(t) \in \mathbb R[t] \}$ for some $v_0 \in V$.

Find all $T$-invariant subspaces of $V$.

Let $W\subseteq V$ be a $T$-invariant subspace. By previous results, recall that the induced endomorphism in the quotient space $\tilde T: V/W \rightarrow V/W$ and the restriction endomorphism $T_W:W \rightarrow W$ are such that $\chi_T = \chi_{T_W} \cdot \chi_{\tilde T}$

For $p \in \mathbb R[t]$, write $V_{p}$ to denote $V_p = \{v\in V: p(T)(v) = 0\}$. Investigating the $T$-invariant subspaces by its dimension (the $0$-dimensional and $4$-dimensional being the trivial ones), the condition $\chi_T = \chi_{T_W} \cdot \chi_{\tilde T}$ gives us the following possibilities

$\chi_{T_W} \in \{ t-2, t\} $ for $W$ 1-dimensional,

$\chi_{T_W} \in \{ (t-2)t, (t-2)^2\}$ for $W$ 2-dimensional,

$\chi_{T_W} \in \{t(t-2)^2, (t-2)^3\}$ for $W$ 3-dimensional.

Now, since $V$ is $T$-cyclic, all its $T$-invariant subspaces are cyclic as well (see Invariant subspace of cyclic space is cyclic). In particular, the minimal polynomial of $T_W$ equals $\chi_{T_W}.$ It seems like this conditions is able to ensure that the $T$-invariant spaces are $V_{t-2}, V_t$ (1-dimensional), $ V_{(t-2)t}, V_{(t-2)^2}$ (2-dimensional) and $V_{t(t-2)^2}, V_{(t-2)^3}$ (3-dimensional), but I'm not sure on how to prove this. Am I going in the right direction?

user2345678
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  • So, $V$ as an $\mathbb{R}[T]$-module is isomorphic to $\mathbb{R}[T] / \langle (T-2)^3 T \rangle$, and it would be equivalent to look at the $\mathbb{R}[T]$-submodules of the latter. I would tend to think in terms of the generator of the submodule: it should be the case that for every $p$, $\langle p \rangle = \langle \gcd(p, (T-2)^3 T) \rangle$ and you're left with submodules $\langle p \rangle$ where $p$ is some (monic) factor of $(T-2)^3 T$. (Haven't checked the details sufficiently to write a full answer.) – Daniel Schepler Feb 04 '21 at 00:11
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    OK, to simplify even further: the $\mathbb{R}[T]$-submodules of $\mathbb{R}[T] / \langle (T-2)^3 T \rangle$ are exactly the same as the ideals of the quotient ring. Then, those ideals are in canonical bijection with the ideals of $\mathbb{R}[T]$ which contain $\langle (T-2)^3 T \rangle$. But each ideal of $\mathbb{R}[T]$ is a principal ideal $\langle p \rangle$ for some $p \in \mathbb{R}[T]$; and $\langle (T-2)^3 T \rangle \subseteq \langle p \rangle$ if and only if $p \mid (T-2)^3 T$. – Daniel Schepler Feb 04 '21 at 00:28
  • I'm not 100% familiar with this language, altough I think that I can keep up (I have never made this exact connection between commutative algebra and linear algebra before). So the submodules of $\mathbb R[T]/\langle (T-2)^3T\rangle $ are all of the form $\mathbb R[T]/\langle p \rangle$? How does that translate into the language of subspaces of $V$? Are they $V_p$? – user2345678 Feb 04 '21 at 00:41
  • The submodule generated by $p$ would correspond in the original language to ${ q(T) p(T) v_0 \mid q \in \mathbb{R}[T] } = { q(T) v_0 \mid q \in \mathbb{R}[T] \land (p \mid q) }$. And then, for instance the dimension of the corresponding vector space would be equal to $4 - \deg(p)$. – Daniel Schepler Feb 04 '21 at 00:46
  • In your notation, the submodule generated by $p$ would correspond to your $V_{t(t-2)^3 / p}$. – Daniel Schepler Feb 04 '21 at 00:48
  • So they are the ones I was expecting them to be. I liked this approach a lot. Would you recommend a book on this subject? I want to make sure I understood everything. – user2345678 Feb 04 '21 at 00:53

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