  For $n\geq 2$, we consider the map on $M_n(\mathbb K)$ given by evaluation of
a polynomial $f(X_1, \ldots, X_m)$ over the field $\mathbb K$. In this article,
we explore the image of the diagonal map given by $f=\delta_1 X_1^{k_1} +
\delta_2 X_2^{k_2} + \cdots +\delta_m X_m^{k_m}$ in terms of the solution of
certain equations over $\mathbb K$. In particular, we show that for $m\geq 2$,
the diagonal map is surjective when (a) $\mathbb K= \mathbb C$, (b) $\mathbb K=
\mathbb F_q$ for large enough $q$. Moreover, when $\mathbb K= \mathbb R$ and
$m=2$ it is surjective except when $n$ is odd, $k_1, k_2$ are both even, and
$\delta _1\delta_2>0$ (in that case the image misses negative scalars), and the
map is surjective for $m\geq 3$. We further show that on $M_n(\mathbb H)$ the
diagonal map is surjective for $m\geq 2$, where $\mathbb H$ is the algebra of
Hamiltonian quaternions.
