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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 $\operatorname{sym}(\mathbb{R}^{n})$.

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

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