  We study the large $N$ matrix model for the index of 4d $\mathcal{N}=4$
Yang-Mills theory and its truncations to understand the dual AdS$_5$ black
holes. Numerical studies of the truncated models provide insights on the black
hole physics, some of which we investigate analytically with the full
Yang-Mills matrix model. In particular, we find many branches of saddle points
which describe the known black hole solutions. We analytically construct the
saddle points dual to the small black holes whose sizes are much smaller than
the AdS radius. They include the asymptotically flat BMPV black holes embedded
in large AdS with novel thermodynamic instabilities.
