  We define a random commuting $d$-tuple of $n$-by-$n$ matrices to be a random
variable that takes values in the set of commuting $d$-tuples and has a
distribution that is a rapidly decaying continuous weight on this algebraic
set. In the Hermitian case, we characterize the eigenvalue distribution as $n$
tends to infinity. In the non-Hermitian case, we get a formula that holds if
the set is irreducible. We show that there are qualitative differences between
the single matrix case and the several commuting matrices case.
