smooth change of basis $p(x)$ such that, egin{align} p^t(x)q(x)p(x) &= egin{pmatrix} id_p & & \ & -id_q & \ & & _r\end{pmatrix} \end{align} my attempt to prove it, assume $q(x_ )$ is invertible and that all eigenvalues are simple. these properties will smoothly depend on the point and then will be preserved...
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