Sum-product estimates for diagonal matrices

Given $d \in \mathbb{N}$, we establish sum-product estimates for finite, non-empty 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, non-empty set of $d \times d$ diagonal matrices with real entries. Then for all $\delta_1 < 1/3 + 5/5277$, we have $$|A+A| + |A\cdot A| \gg_{d} |A|^{1 + \delta_{1}/d}.$$ In this setting, the above estimate quantitatively strengthens a result of Chang.