Symmetric matrices with distinct eigenvalues are open and dense

Idea. The spectral theorem lets us change the eigenvalues of a symmetric matrix at will without leaving $\mathrm{Sym}_n$: we keep the orthogonal factor and move the diagonal one.

Let $\mathcal G=\{S\in\mathrm{Sym}_n:S\text{ has }n\text{ distinct eigenvalues}\}$.

Step 1: $\mathcal G$ is open in $\mathrm{Sym}_n$. $\mathcal G=\{S\in\mathrm{Sym}_n:\Delta(S)\ne0\}$ where $\Delta(S)=\operatorname{disc}(\chi_S)$ is a polynomial in the entries (previous problems). It is the preimage of $\mathbb{R}\setminus\{0\}$ under a continuous map restricted to $\mathrm{Sym}_n$, hence open in $\mathrm{Sym}_n$.

Step 2: $\mathcal G$ is dense in $\mathrm{Sym}_n$. Let $S\in\mathrm{Sym}_n$. By the spectral theorem $S=Q\Lambda Q^T$ with $Q$ orthogonal and $\Lambda=\operatorname{diag}(\lambda_1,\dots,\lambda_n)$. For $\varepsilon\ne0$ put
$$S_\varepsilon=Q\big(\Lambda+\varepsilon\operatorname{diag}(1,2,\dots,n)\big)Q^T.$$
This is symmetric (it is $Q\,\mathrm{diag}\,Q^T$), and its eigenvalues are $\lambda_i+i\varepsilon$. These are pairwise distinct unless $\lambda_i+i\varepsilon=\lambda_j+j\varepsilon$ for some $i<j$, i.e. $\varepsilon=\frac{\lambda_i-\lambda_j}{j-i}$: finitely many values of $\varepsilon$ to avoid. So for all small $\varepsilon\ne0$, $S_\varepsilon\in\mathcal G$. Finally, orthogonal invariance of the operator norm gives
$$\|S_\varepsilon-S\|=\varepsilon\,\|\operatorname{diag}(1,\dots,n)\|=n\varepsilon\to0.$$

Remark.
- Both statements together say that *a generic symmetric matrix has simple spectrum*, which is why proofs for symmetric matrices can usually assume distinct eigenvalues and conclude by continuity.
- The complement of $\mathcal G$ in $\mathrm{Sym}_n$ is closed with empty interior, but unlike in $M_n(\mathbb{R})$ it has codimension $2$, not $1$: two eigenvalues of a symmetric matrix collide in codimension $2$ (von Neumann-Wigner "level repulsion"), e.g. in $\mathrm{Sym}_2\cong\mathbb{R}^3$ the repeated-eigenvalue matrices are the *line* of scalar matrices $\{aI\}$.