Multiply partition regular matrices

Let AA be a finite matrix with rational entries. We say that AA is doubly image partition regular   if whenever the set NN of positive integers is finitely coloured, there exists x→ such that the entries of Ax→ are all the same colour (or monochromatic  ) and also, the entries of x→ are monochromatic. Which matrices are doubly image partition regular?More generally, we say that a pair of matrices (A,B)(A,B), where AA and BB have the same number of rows, is doubly kernel partition regular   if whenever NN is finitely coloured, there exist vectors x→ and y→, each monochromatic, such that Ax→+By→=0→. (So the case above is the case when BB is the negative of the identity matrix.) There is an obvious sufficient condition for the pair (A,B)(A,B) to be doubly kernel partition regular, namely that there exists a positive rational cc such that the matrix M=(AcB)  is kernel partition regular. (That is, whenever NN is finitely coloured, there exists monochromatic x→ such that Mx→=0→.) Our aim in this paper is to show that this sufficient condition is also necessary. As a consequence we have that a matrix AA is doubly image partition regular if and only if there is a positive rational cc such that the matrix (A−cI)  is kernel partition regular, where II is the identity matrix of the appropriate size.We also prove extensions to the case of several matrices.