Consider the generalized eigenvalue problem
$$A v = \lambda B v$$
where $v \in \mathbb{R}^n, A, B \in \mathbb{R}^{n \times n}$.
Suppose that $A, B$ are symmetric and positive semi-definite. Suppose further that $B$ is positive definite (if it matters, because it is of the form $\alpha I + (1-\alpha)C$ for $\alpha \in (0,1)$ and $C\in \mathbb{R}^{n \times n}$ symmetric and positive semi-definite).
Suppose $\text{rank}(A) = m < n$.
What can be said about the solutions to this eigenvalue problem? (e.g.: about eigenspaces, smallest eigenvalues, etc.)
In particular, what can be said about the number of distinct generalized eigenvectors? Under what conditions (if any) is this number $m$?
I am mainly interested in those questions, but I'd also really appreciate any general resources about these kinds of problems that may give me any structural results at all.
Thanks!