  This paper extends the Bakry-\'{E}mery theorem connecting the Ricci curvature
and log-Sobolev inequalities to the matrix-valued setting. Using tools from
noncommuative geometry, it is shown that for a right invariant second order
differential operator on a compact Lie group, a lower bound for a matrix-valued
modified log-Sobolev inequality is equivalent to a uniform lower bound for all
finite dimensional representations. Using combinatorial tools, we obtain
computable lower bounds for matrix-valued log-Sobolev inequalities of
graph-H\"ormander systems using combinatorial methods.
