So it's a vector-valued linear differential equation.

The solution is a natural generalization from the case where \$$D\$$ contains just one element. In that case, the linear

\$H: \mathbb{R}^1 \rightarrow \mathbb{R}^1\$

amounts to multiplication by a constant \$$H\$$, and the master equation takes the simple form:

\$\Gamma'(t) = H \cdot \Gamma(t)\$

which is an elementary differential equation, with solution:

\$\Gamma(t) = e^{H t}\ \Gamma(0)\$