  Real Nullstellensatz is a classical result from Real Algebraic Geometry. It
has recently been extended to quaternionic polynomials by Alon and Paran. The
aim of this paper is to extend their Quaternionic Nullstellensatz to matrix
polynomials. We also obtain an improvement of the Real Nullstellensatz for
matrix polynomials in the sense that we simplify the definition of a real left
ideal. We use the methods from the proof of the matrix version of Hilbert's
Nullstellensatz and we obtain their extensions to a mildly non-commutative case
and to the real case.
