I would like to know which are the invariant subspaces for the endomorphisms $f1$, $f2$, $f3$, $f4$, $f5$ from vector space $V$ that have the next associated Jordan matrices:
$J1 = \left( \begin{array}{cccc} -1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 2 & 1\\ 0 & 0 & 0 & 2 \end{array} \right)$
$J2 = \left( \begin{array}{cccc} -1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 2 \end{array} \right)$
$J3 = \left( \begin{array}{cccc} 1 & 1 & 0 & 0\\ 0 & 1 & 1 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array} \right)$
$J4 = \left( \begin{array}{cccc} -1 & 1 & 0 & 0\\ 0 & -1 & 1 & 0\\ 0 & 0 & -1 & 1\\ 0 & 0 & 0 & -1 \end{array} \right)$
$J5 = \left( \begin{array}{cccc} 3 & 1 & 0 & 0\\ 0 & 3 & 0 & 0\\ 0 & 0 & 3 & 1\\ 0 & 0 & 0 & 3 \end{array} \right)$
Thank you very much.
The space spanned by the generalized eigenvectors associated with each simple Jordan block is invariant with respect to the relevant endomorphism. So that is if you decompose $V$ into a direct sum of the smallest possible invariant spaces (with respect to the relevant endomorphism). Now you also have that each generalized eigenspace associated with a specific eigenvalue is also invariant with respect to the relevant endomorphism.
So let's consider an example - let's take the first endomorphism $f_1$ and its Jordan form $J_1$. Then we can express $V$ as a direct sum of $f_1$-invariant subspaces: \begin{equation} V=V_{-1,1}\oplus V_{-1,2} \oplus V_{2,1}, \end{equation} where $V_{\lambda,i}$ represents the subspace spanned by the generalized eigenvectors of $f_1$ associated with each simple block $J_i(\lambda)$. Now we also have $E_{f_1}(-1)=V_{-1,1}\oplus V_{-1,2}$ is the generalized eigenspace associated with eigenvalue -1, and $E_{f_1}(2)=V_{2,1}$ is the generalized eigenspace associated with eigenvalue 2. So we also have $V$ expressed as a direct sum of $f_1$-invariant subspaces: \begin{equation} V=E_{f_1}(-1) \oplus E_{f_1}(2). \end{equation}
I hope that helps...you should be able to do the others on your own?