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A paper I'm reading in representation theory states the following result:

Let $F$ be a field of characteristic zero, and $x$ a symmetric matrix in $M_n(F)$ all of whose principal minors are not zero. Then there exists an upper triangular unipotent matrix $u$ such that $ux \space ^tu$ is diagonal.

I am looking for a reference for this result, or a suggestion on how to think about this to be able to prove it myself.

With respect to the standard basis in $F^n$, symmetric matrices correspond to symmetric bilinear forms, with invertible matrices corresponding to nondegenerate forms. I am trying to think about how to interpret the fact that the principal subdeterminants of $x$ are also nonzero, in terms of the symmetric form.

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  • $\begingroup$ As far as I remember, this should bear the name `Jacobi's method'... $\endgroup$ – Ilya Bogdanov Oct 15 '17 at 13:11
  • $\begingroup$ anyway, this is what happens in the method here: math.stackexchange.com/questions/1388421/… when Case I always applies. $\endgroup$ – Will Jagy Oct 15 '17 at 18:18
  • $\begingroup$ hmmm From Horn and Johnson, it seems Jacobi's method is rather specific, conerging to eigenvalues on the diagonal...... $\endgroup$ – Will Jagy Oct 15 '17 at 18:31

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