Solution
= Solution
For $D=P^{-1}D_0P$, conjugating $U=D^{-1}T$ gives
$$
PUP^{-1}=D_0^{-1}(D_0+N)=I+D_0^{-1}N.
$$
Because $D_0^{-1}$ commutes with $N$, and $N^n=0$,
$$
(D_0^{-1}N)^n=D_0^{-n}N^n=0.
$$
Thus $U-I$ is similar to a <nilpotent matrix>, and is itself nilpotent. \b[Therefore $U$ is a unipotent matrix.]