# Orbit closures of real symmetric bilinear forms

Let $$\alpha$$ and $$\beta$$ be two real symmetric bilinear forms in $$\operatorname{sym}(\mathbb{R}^{n})$$, with signatures $$(p_{\alpha},n_{\alpha},z_{\alpha})$$ and $$(p_{\beta},n_{\beta},z_{\beta})$$.

Please, I would like to have some references or bibliography about published papers concerning to the following theorem:

Theorem $$β∈\overline{ \operatorname{GL}(n,\mathbb{R})⋅α}$$ if and only if $$p_{\alpha}\geq p_{\beta}$$ and $$n_{\alpha}\geq n_{\beta}$$.

Here, $$\operatorname{GL}(n,\mathbb{R})⋅\alpha:=\{\alpha(g^{−1}⋅,g^{−1}⋅):g\in \operatorname{GL}(n,\mathbb{R})\}$$ and $$\overline{ \operatorname{GL}(n,\mathbb{R})⋅α}$$ is the closure of $$\operatorname{GL}(n,\mathbb{R})⋅α$$ with respect to the usual (Euclidean) topology of $$\operatorname{sym}(\mathbb{R}^{n})$$.

• It doesn't really answers your request for references, but isn't it an immediate consequence of Sylvester's law of inertia? – sss89 Oct 2 at 15:37