  In this paper a general theory of semi-classical matrix orthogonal
polynomials is developed. We define the semi-classical linear functionals by
means of a distributional equation $D(u A) = u B,$ where $A$ and $B$ are matrix
polynomials. Several characterizations for these semi-classical functionals are
given in terms of the corresponding (left) matrix orthogonal polynomials
sequence. They involve a quasi-orthogonality property for their derivatives, a
structure relation and a second order differo-differential equation. Finally we
illustrate the preceding results with some non-trivial examples.
