  We derive a matrix version of Li \& Yau--type estimates for positive
solutions of semilinear heat equations on Riemannian manifolds with nonnegative
sectional curvatures and parallel Ricci tensor, similarly to what R.~Hamilton
did in~\cite{hamilton7} for the standard heat equation. We then apply these
estimates to obtain some Harnack--type inequalities, which give local bounds on
the solutions in terms of the geometric quantities involved.
