I got stuck in this problem from Spring 99, Berkeley Problems in Mathematics:
Let $A$ be a $n\times n$ matrix such that $a_{ij}\not=0$ if $i=j+1$ but $a_{ij}=0$ if $i\ge j+2$. Prove that $A$ cannot have more than one Jordan block for any eigenvalue.
I thought the matrix would satisfy some obvious relationship like $A^{2}=0$, but I realized the entries not listed are not even specified; thus such a gross simplification cannot hold. Working on toy examples does not tell me much, so I decided to ask in here.