  A sum rule relative to a reference measure on R is a relationship between the
reversed Kullback-Leibler divergence of a positive measure on R and some
non-linear functional built on spectral elements related to this measure (see
for example Killip and Simon 2003). In this paper, using only probabilistic
tools of large deviations, we extend the sum rules obtained in Gamboa, Nagel
and Rouault (2015) to the case of Hermitian matrix-valued measures. We recover
the earlier result of Damanik, Killip and Simon (2010) when the reference
measure is the (matrix-valued) semicircle law and obtain a new sum rule when
the reference measure is the (matrix-valued) Marchenko-Pastur law.
