  We show that any scalar differential operator with a family of polyno- mials
as its common eigenfunctions leads canonically to a matrix differen- tial
operator with the same property. The construction of the correspond- ing family
of matrix valued polynomials has been studied in [D1, D2, DV] but the existence
of a differential operator having them as common eigen- functions had not been
considered This correspondence goes only one way and most matrix valued
situations do not arise in this fashion.
  We illustrate this general construction with a few examples. In the case of
some families of scalar valued polynomials introduced in [GH] we take a first
look at the algebra of all matrix differential operators that share these
common eigenfunctions and uncover a number of phenomena that are new to the
matrix valued case.
