I just want to check that my understanding is correct of invariant subspaces. I was given a matrix A in which I have found that it is invertible, so I know that a $2$-dimensional invariant subspace will have the form of:
$$W = \left\{ \alpha w_{1} + \beta w_{2} \mid \alpha, \beta \in \mathbb{R} \right\}$$
For this just let A be: $$\begin{bmatrix} a &b&c&d\\e&f&g&h\\i&j&k&l\\m&n&o&p \end{bmatrix}$$ And because the matrix is invertible we that $Aw_1$ and $Aw_2$ would span $W$ itself. But to find the invariant subspaces I have changed the matrix into Jordan form and the canonical jordan form is: $$J = \begin{bmatrix} -2&0&0&0\\0&5&0&0\\0&0&-0.37 & 0\\ 0&0&0&0.37\end{bmatrix}$$ Using the Jordan form of the matrix means that each Jordan block are a basis for a flag of invariant subspaces. So what do the subspaces become?
Is my understanding correct?
You are right. The eigenvector correspond to each eigenvalue is a 1d invariant subspace. Every combination (direct sum) of invariant subspaces will still be an invariant subspace. So in your 4 vectors, the subspace spanned by any two of them will be an invariant subspace.
Your two are invariant subspace too .