Number of Invariant Subspaces of a Jordan Block I'm asking this question on behalf of a person I'm supposed to be tutoring who has this problem as part of eir homework.
The problem is "How many invariant subspaces are there of a transformation $T$ that sends $v\mapsto J_{\lambda,n}v$" where $J_{\lambda,n}$ is a Jordan block. We are pretty sure the answer is $n+1$, where the spaces are the trivial space and the ones spanned by sets of columns of this form:
$\Big\{$ 
$\pmatrix{1 \\ 0 \\ 0 \\ \vdots \\ 0 \\ 0 \\ \vdots \\ 0}$ ,
$~\pmatrix{0 \\ 1 \\ 0 \\ \vdots \\ 0 \\ 0 \\ \vdots \\ 0}$ ,
$~\pmatrix{0 \\ 0 \\ 1 \\ \vdots \\ 0 \\ 0 \\ \vdots \\ 0}$ ,
$~\cdots$ ,
$~\pmatrix{0 \\ 0 \\ 0 \\ \vdots \\ 1 \\ 0 \\ \vdots \\ 0}$ 
$\Big\}$ 
.
But we are not sure how to explain that there aren't others. Can anyone help us, or at least put us on the right track?

You are right. Let $J_{\lambda,n}=\lambda\,I+N$ where $N$ is the nilpotent part, which maps $e_i\mapsto e_{i-1}$ if $i>1$ and $e_1\mapsto 0$.


*

*Observe that the invariant subspaces of $\lambda\,I+N$ coincide with those of $N$.

*Assume that $v=(v_1,..,v_k,0,..,0)$ with $v_k\ne 0$, and consider its generated $N$-invariant subspace $V$. We have $V\ni N^{k-1}v=v_k\,e_1$, so $e_1\in V$. Similarly, by $N^{k-2}v,e_1\in V$ we can conclude $e_2\in V$, and so on, until $e_k\in V$. Thus, indeed $V={\rm span}(e_1,..,e_k)$.

