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.