  The Entropy method provides a powerful framework for proving scalar
concentration inequalities by establishing functional inequalities like
Poincare and log-Sobolev inequalities. These inequalities are especially useful
for deriving concentration for dependent distributions coming from stationary
distributions of Markov chains. In contrast to the scalar case, the study of
matrix concentration inequalities is fairly recent and there does not seem to
be a general framework for proving matrix concentration inequalities using the
Entropy method. In this paper, we initiate the study of Matrix valued
functional inequalities and show how matrix valued functional inequalities can
be used to establish a kind of iterative variance comparison argument to prove
Matrix Bernstein like inequalities. Strong Rayleigh distributions are a class
of distributions which satisfy a strong form of negative dependence properties.
We show that scalar Poincare inequalities recently established for a flip-swap
markov chain considered in the work of Hermon and Salez [HS19] can be lifted to
prove matrix valued Poincare inequalities. This can then be combined with the
iterative variance argument to prove a version of matrix Bernstein inequality
for Lipshitz matrix-valued functions for Strong Rayleigh measures.
