A linear map $T: \mathbb{R^3 \to \mathbb{R^3}}$ has a two dimensional invariant subspace.

Question

Let $T: \mathbb{R^3 \to \mathbb{R^3}}$ be an $\mathbb{R}$-linear map. Then I want to show that $T$ has a $2$ dimensional invariant subspace of $\mathbb{R^3}.$

I considered all possible minimal polynomial of $T$ and applying canonical forms I found some obvious $2$ dimensional invariant subspaces.

I stuck when the minimal polynomial is of the form $(X-a)^3$ for some real number $a.$

In this situation since the minimal polynomial and the characteristic polynomial coincides $T$ has a cyclic vector. But I can't complete it further. I need some help. Thanks.

Answer 1

If the minimal polynomial is of the form $(X-a)^3$, then there is some basis $B=\{e_1,e_2,e_3\}$ of $\mathbb{R}^3$ such that the matrix of $T$ with respect to $B$ is$$\begin{bmatrix}a&1&0\\0&a&1\\0&0&a\end{bmatrix}.$$So, consider the space spanned by $e_1$ and $e_2$.