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Consider a matrix $A\in\mathbb{R}^{n\times n}$ and a projector $P\in\mathbb{R}^{n\times n}$.

Are there results regarding the generalized eigenpairs $(v,\lambda)$ of the generalized eigenproblem

$$ Av = \lambda Pv, $$ e.g., characterizations, bounds depending on the range of $P$, etc?

I can derive some things, but I don't want to reinvent the wheel. I did some searching, but any search with "generalized eigen.." and "projection" pulls numerical techniques on that topic. I am wondering if perhaps this problem has a different name that I can reference for a proper search.

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