Solution
= Solution
If $U$ is a <unipotent matrix>, $N=U-I$ is a <nilpotent matrix>. If $Uv=\lambda v$ for a nonzero <eigenvector>, then $N^mv=(\lambda-1)^mv=0$ for some $m$, forcing $\lambda=1$.
Conversely, suppose the only <eigenvalue> is $1$. Over $\mathbb C$, the <characteristic polynomial> is then $(t-1)^n$. The <Cayley-Hamilton theorem> gives $(U-I)^n=0$, so $U-I$ is nilpotent. \b[Thus $U$ is unipotent if and only if its only eigenvalue is one.]