From "Oscillations" chapter in Classical mechanics Goldstein
$A=H^{-1}U$,
then the eigenvalue of A,
$\lambda = \frac{x^{T}Ux}{x^{T}Hx}$
where $x$ is the eigenvector of $A$ and U, H are real symmetric matrices.
If $y$ is the eigenvector of $U$, then what is the relationship between $y$ and $x$?
P.S: I am looking to prove that when the eigenvalues of U increases, corresponding $\lambda $ also increases.
Is it possible to derive a relationship such as $y^{T}Uy=k.x^{T}Ux$? Then I could show that if LHS increases, then RHS also increases.