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I have the following generalized eigenvalue problem from classical mechanics

$$\omega^2 Tv = Uv$$

where $T$ is a real, positive definite matrix and $U$ is positive semidefinite. To find the eigenvalues, I have to calculate

$$\det(-\omega^2 T + U) = 0$$

Why can't I simply write $T^{-1}Uv = \omega^2 v$ and solve it as a standard eigenvalue problem?

Can you suggest me any resources (books, etc) about this topic?

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