Given
$d\in \mathbb{N}$
, we establish sum-product estimates for finite, nonempty subsets of
$\mathbb{R}^{d}$
. This is equivalent to a sum-product result for sets of diagonal matrices. In particular, let
$A$
be a finite, nonempty set of
$d\times d$
diagonal matrices with real entries. Then, for all
$\unicode[STIX]{x1D6FF}_{1}<1/3+5/5277$
,
$$\begin{eqnarray}|A+A|+|A\cdot A|\gg _{d}|A|^{1+\unicode[STIX]{x1D6FF}_{1}/d},\end{eqnarray}$$
which strengthens a result of Chang [‘Additive and multiplicative structure in matrix spaces’,
Combin. Probab. Comput.16(2) (2007), 219–238] in this setting.