# 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?

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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 $e_1\mapsto 0$.
1. Observe that the invariant subspaces of $\lambda\,I+N$ coincide with those of $N$.
2. 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 $V={\rm span}(e_1,..,e_k)$.