The resultant for regular matrix polynomials and quasi commutativity

Necessary and sufficient conditions are presented in order that for two regular matrix polynomials the null space of the straightforward analogue of the classical Sylvester resultant matrix is completely determined by the common spectral data of the polynomials involved. The conditions are stated in terms of a quasi commutativity property. In the scalar case, and more generally in the case of commuting matrix polynomials, the conditions are automatically satisfied.