PSD matrices have empty interior in all of M_n

Idea. PSD matrices are symmetric by definition, and $\mathrm{Sym}_n$ is a proper subspace of $M_n(\mathbb{R})$ when $n\ge2$. A subset of a set with empty interior has empty interior.

Step 1: PSD $\subset\mathrm{Sym}_n$ and $\mathrm{Sym}_n\ne M_n$. For $n\ge2$ the matrix $E_{12}$ is not symmetric, so $\mathrm{Sym}_n$ is a *proper* linear subspace.

Step 2: $\mathrm{Sym}_n$ has empty interior. This is the earlier result on proper subspaces. Explicitly (the skew-symmetric perturbation of the hint): let $A\in\mathrm{Sym}_n$ and let $K=E_{12}-E_{21}$ (skew-symmetric, $\|K\|=1$). Then $A+\varepsilon K\notin\mathrm{Sym}_n$ for every $\varepsilon\neq0$ (its transpose is $A-\varepsilon K$), and $\|A+\varepsilon K-A\|=\varepsilon$. So every ball around $A$ contains a non-symmetric matrix.

Step 3: conclusion. Any ball around a PSD matrix contains a non-symmetric matrix, hence a matrix outside PSD. So $\operatorname{int}_{M_n}(\mathrm{PSD})=\emptyset$.

Remark.
- For $n=1$ the claim is false: $M_1=\mathbb{R}$ and $\mathrm{PSD}=[0,\infty)$ has interior $(0,\infty)$.
- Be careful with definitions. If "positive semidefinite" is defined for *non-symmetric* real matrices by the condition $x^TAx\ge0$ for all $x$, the set is $\mathrm{PSD}+\mathrm{Skew}_n$ and does have non-empty interior (e.g. $I+K$ for small skew $K$). In this book PSD means *symmetric* with $x^TAx\ge0$.
- This is why one always says "interior in $\mathrm{Sym}_n$": the ambient space matters for interior and boundary.