# A symmetric matrix with nonzero principal minors is cogredient to a diagonal matrix via an upper triangular

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.

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