Distance Between Symmetric Real Matrices

Let $n\ge1$ be a natural number, let $\mathbb{R}$ be the real numbers, and let $\mathcal{S}(n)$ be the set of symmetric real $n\times n$ matrices. For $X,Y\in\mathcal{S}(n)$ the difference $X-Y$ is again symmetric by claim 1 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure, hence lies in $\mathcal{S}(n)$ and has a norm $\lVert X-Y\rVert$.

The distance between symmetric real $n\times n$ matrices is the function

$$d_{\mathcal{S}(n)}:\mathcal{S}(n)\times\mathcal{S}(n)\to\mathbb{R}$$

given by

$$d_{\mathcal{S}(n)}(X,Y)=\lVert X-Y\rVert .$$