# What do we know about the generalized eigenvalue problem involving a projector?

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.

• Have you looked for 'matrix pencils'? en.wikipedia.org/wiki/Matrix_pencil Not that I have something specific to say about your projection case, but it might help to your search. Apr 15 '14 at 13:07