  In this paper we prove some characterizations of the matrix orthogonal
polynomials whose derivatives are also orthogonal, which generalize other known
ones in the scalar case. In particular, we prove that the corresponding
orthogonality matrix functional is characterized by a Pearson-type equation
with two matrix polynomials of degree not greater than 2 and 1. The proofs are
given for a general sequence of matrix orthogonal polynomials, not necessarily
associated with an hermitian functional. However, we give several examples of
non-diagonalizable positive definite weight matrices satisfying a Pearson-type
equation, which show that the previous results are non-trivial even in the
positive definite case.
  A detailed analysis is made for the class of matrix functionals which satisfy
a Pearson-type equation whose polynomial of degree not greater than 2 is
scalar. We characterize the Pearson-type equations of this kind that yield a
sequence of matrix orthogonal polynomials, and we prove that these matrix
orthogonal polynomials satisfy a second order differential equation even in the
non-hermitian case. Finally, we prove and improve a conjecture of Duran and
Grunbaum concerning the triviality of this class in the positive definite case,
while some examples show the non-triviality for hermitian functionals which are
not positive definite.
