Jordan matrices with the only eigenvalue $1$.

Question

So, I need to list all of the Jordan matrices of a $4x4$ matrix with the only eigenvalue $1$. If the only eigenvalue is 1, then there can't be any other value on the diagonal, correct? So am I wrong in thinking that only Jordan matrix would be this? $$ J_{1,n} =\begin{bmatrix} 1 & 1 & 0 & 0 \\ 0&1&1&0\\0&0&1&1\\0&0&0&1 \end{bmatrix}$$

Answer 1

Read egreg's comment and complete. Some of the possibilities are:

$$\begin{align}&4=4\;\;\rightarrow&\begin{pmatrix}1&1&0&0\\0&1&1&0\\0&0&1&1\\0&0&0&1\end{pmatrix}\\{}\\ &4=3+1\;\;\rightarrow&\begin{pmatrix}1&1&0&0\\0&1&1&0\\0&0&1&0\\0&0&0&1\end{pmatrix}\\{}\\ &4=2+2\;\;\rightarrow&\begin{pmatrix}1&1&0&0\\0&1&0&0\\0&0&1&1\\0&0&0&1\end{pmatrix}\end{align}$$

Can you see the relation? $\;4=n+k+r=$ one block of size $\;n\;$ and one of size $\;k\;$ and one of size $\;r\;$ (and etc.), and thus, by the comment quoted before, there are as many possibilities as partitions of $\;4\;$ , which are five.

Complete now according to the other partitions.