I've achived to find invariant subspaces only once, but today I've got a problem to solve involving invariant spaces, and I'm not sure how to procedeed. Indeed I would like to find the invariant subspaces of the linear endomorphism of the coordinate space $F^n$ given by the following Jordan block:
$$\left(\begin{array}{ccccccc} \lambda & 1 & 0 & \cdots & 0 & 0 & 0\\ 0 & \lambda & 1 & \cdots & 0 & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \\ 0 & 0 & 0 & \cdots & \lambda & 1 & 0\\ 0 & 0 & 0 & \cdots & 0 & \lambda & 1\\ 0 & 0 & 0 & \cdots & 0 & 0 & \lambda \end{array}\right)$$
Could anyone explain to me a little bit how to invariant subspaces in this case ? Thanks in advance for your time.
Hint: letting $M$ denote your matrix, and writing $T(x) = Mx$ (i.e., multiplying on the right by a column vector $x$), we see that
$$ \operatorname{span}(e_1) $$ is invariant, because $T(e_1) = \lambda e_1 \in Span(e_1)$, where $e_1$ is the vector $$ \pmatrix{ 1\\0\\ \vdots\\0}. $$
What about $\operatorname{span}(e_1, e_2)$?