  A general family of matrix valued Hermite type orthogonal polynomials is
introduced and studied in detail by deriving Pearson equations for the weight
and matrix valued differential equations for these matrix polynomials. This is
used to derive Rodrigues formulas, explicit formulas for the squared norm and
to give an explicit expression of the matrix entries as well to derive a
connection formula for the matrix polynomials of Hermite type. We derive matrix
valued analogues of Burchnall formulas in operational form as well explicit
expansions for the matrix valued Hermite type orthogonal polynomials as well as
for previously introduced matrix valued Gegenbauer type orthogonal polynomials.
The Burchnall approach gives two descriptions of the matrix valued orthogonal
polynomials for the Toda modification of the matrix weight for the Hermite
setting. In particular, we obtain a non-trivial solution to the non-abelian
Toda lattice equations.
