For \(n\geqslant 2\) , we consider the polynomial maps on \(M_n(\mathbb K)\) given by evaluation of a polynomial \(f(X_1, \ldots , X_m)\) over the field \(\mathbb K\) . We explore the image of the diagonal map given by in terms of the solution of certain equations over \(\mathbb K\) . We show that when \(\mathbb K= \mathbb R\) and \(m=2\) , it is surjective except when n is odd, \(\delta _1\delta _2>0\) , and \(k_1, k_2\) are both even (in that case, the image misses negative scalars), and the map is surjective for \(m\geqslant 3\) . We further show that on \(M_n(\mathbb H)\) (even with \(\mathbb H\) coefficients) the diagonal map is surjective for \(m\geqslant 2\) , where \(\mathbb H\) is the algebra of Hamilton’s quaternions.
