The Set of Symmetric Real Matrices is a Metric Space

Let $n\ge1$ be a natural number, let $\mathcal{S}(n)$ be the set of symmetric real $n\times n$ matrices, and let $d_{\mathcal{S}(n)}$ be the distance between symmetric real matrices.

Then $d_{\mathcal{S}(n)}$ is a metric on $\mathcal{S}(n)$, so that $\bigl(\mathcal{S}(n),d_{\mathcal{S}(n)}\bigr)$ is a metric space.