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In my mulitivariable calculus class to justify second derivative test my professor used a theorem he called the frobenius theorem. But when I searched on wiki all I could find was Perron Frobenius Theorem. So I was unable to find the proof of the theorem.

The theorem is

All eigen values of a real symmetric matrix are positive if $A_1,A_2,...,A_n $ are positive where $A_m$=det $([a_{ij}]_{m \times m})$

I tried to prove this statement first using Induction in the following way: First I did a change of basis transformation a make one of the basis vectors into an eigen vector and then I wanted to apply induction. But when I do a change of basis I was not able to prove that the determinants stay the same. So I could not go ahead.

Then I tried to use Gaussian Elimination which preserves determinants but I am not sure it preserves Eigen values. So I am struck again. Can anyone give any hints as to how to solve this. Thanks.

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    $\begingroup$ This is usually known as Sylvester's criterion in textbooks. $\endgroup$ – user1551 Feb 2 '15 at 3:52
  • $\begingroup$ @user1551 thanks a lot for the reference I was searching for exactly this $\endgroup$ – happymath Feb 2 '15 at 3:54

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