Components of the invertible symmetric matrices

Idea. The signature is a continuous integer-valued invariant (the eigenvalues cannot cross $0$ without making the matrix singular), and matrices with the same signature are connected through the spectral theorem.

Step 1: continuity of eigenvalues. Let $\mathrm{Sym}_n^*$ be the set of invertible symmetric matrices and $\lambda_1(S)\ge\dots\ge\lambda_n(S)$ the ordered eigenvalues of $S$. By *Weyl's inequality*,
$$|\lambda_i(S)-\lambda_i(S')|\le\|S-S'\|\quad(\text{operator norm}),$$
so each $\lambda_i$ is continuous (even $1$-Lipschitz) on $\mathrm{Sym}_n$.

Step 2: the classes are clopen. On $\mathrm{Sym}_n^*$ no $\lambda_i$ vanishes. The number of positive eigenvalues is $k(S)=\#\{i:\lambda_i(S)>0\}$. Since each $\lambda_i$ is continuous and never $0$ on $\mathrm{Sym}_n^*$, the sign of each $\lambda_i$ is locally constant, hence so is $k$. So the classes $\mathcal{C}_k=\{S\in\mathrm{Sym}_n^*:k(S)=k\}$, $k=0,\dots,n$, are open, disjoint, non-empty (they contain $\operatorname{diag}(I_k,-I_{n-k})$) and cover $\mathrm{Sym}_n^*$; each is also closed in $\mathrm{Sym}_n^*$ as a complement of the union of the others.

Step 3: each class is connected. By the spectral theorem every $S\in\mathcal{C}_k$ is $S=Q\operatorname{diag}(d)Q^T$ with $Q\in O(n)$ and $d\in\Delta_k=(0,\infty)^k\times(-\infty,0)^{n-k}$.
- *We may take $Q\in SO(n)$.* Let $D'=\operatorname{diag}(-1,1,\dots,1)$. Since $D'\operatorname{diag}(d)D'=\operatorname{diag}(d)$, $S=(QD')\operatorname{diag}(d)(QD')^T$ and one of $Q,QD'$ has determinant $1$.
- *Connected parameter space.* $SO(n)$ is path connected (Problem 16) and $\Delta_k$ is convex. So $SO(n)\times\Delta_k$ is path connected, and $\mathcal{C}_k$ is its continuous image under $(Q,d)\mapsto Q\operatorname{diag}(d)Q^T$.

Conclusion. The path components are exactly $\mathcal{C}_0,\dots,\mathcal{C}_n$: $n+1$ of them.

Remark. $\mathcal{C}_n$ is the cone of positive definite matrices and $\mathcal{C}_0$ its negative. $\mathcal{C}_k$ for $0<k<n$ are the sets of *quadratic forms of signature $(k,n-k)$*, in the classification of quadratic forms by Sylvester's law of inertia. For $n=2$ there are three components: the two cones $\{ac>b^2,a>0\}$, $\{ac>b^2,a<0\}$ and the "inside of the light cone" $\{ac<b^2\}$.