  We show that any probability measure satisfying a Matrix Poincar\'e
inequality with respect to some reversible Markov generator satisfies an
exponential matrix concentration inequality depending on the associated matrix
carr\'e du champ operator. This extends to the matrix setting a classical
phenomenon in the scalar case. Moreover, the proof gives rise to new matrix
trace inequalities which could be of independent interest. We then apply this
general fact by establishing matrix Poincar\'{e} inequalities to derive matrix
concentration inequalities for Gaussian measures, product measures and for
Strong Rayleigh measures. The latter represents the first instance of matrix
concentration for general matrix functions of negatively dependent random
variables.
