I have a $17\times 17$ idempotent matrix $M$ with rank $10$
$M$ is idempotent so $M^2=M \rightarrow M(M-1)=0$. So the minimum polynomial is $x(x-1)$. The eigenvalues of $M$ are $x=0$ and $x=1$.
The rank of $M$ is $10$, so $\dim \ker M=7$.
But I'm stuck determining the number of blocks and the sizes of the blocks of the Jordan normal form.