  Let $c_q(n)$ be the Ramanujan sums and let $S(m,n;q)$ be the Kloosterman
sums. We study the eigenvalues of $q \times q$ matrices whose $(m,n)$ entry is
$c_q (m-n)$ or $S(m,n;q)$ where $q$ is a fixed positive integer. We also study
the eigenvalues of matrices whose entries are sums of Ramanujan sums or sums of
Kloosterman sums.
