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I have been asked to find all invariant subspaces of the linear map given by the Jordan Block matrix: $$J_{2}(\lambda)\in\mathbb{R}^{2x2}, \quad\lambda\in\mathbb{R}$$

I know that $$J_{2}(\lambda)=\left( \begin{array}{cc} \lambda & 1 \\ 0 & \lambda \end{array} \right)$$ but I don't clearly understand what I'm being asked to do. How would you solve this problem?

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You can think of $J_2(\lambda)$ as a map $J_2(\lambda):\mathbb{R}^2 \rightarrow \mathbb{R}^2$. An invariant subspace is some subspace $U \subset \mathbb{R}^2$ such that $J_2(\lambda)(U) \subset U$, so that $J_2(\lambda)$ restricts to a map $J_2(\lambda):U \rightarrow U$. $\{0\}$ and the whole space $\mathbb{R}^2$ are always invariant, and all other subspaces are one-dimensional, so you need to see if there are any one-dimensional invariant subspaces. This is equivalent to looking for eigenvectors, since an eigenvector by definition $(Tv=\lambda v, v \neq 0, \lambda \in \mathbb{R})$ is a vector which spans a one-dimensional invariant subspace.

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