Problem
Let's say I can write a model as the Hermitian eigensystem: $$ A x = \lambda x $$ where $A \in \mathbb{C}^{n\times n}$ is Hermitian, or as the generalized Hermitian eigensystem: $$ \tilde A \tilde x = \lambda \tilde B \tilde x $$ where $\tilde A, \tilde B \in \mathbb{C}^{\tilde n \times \tilde n}$ are Hermitian and $\tilde B$ is positive definite. How much smaller must $\tilde n$ be than $n$ for the generalized model to be more efficient?
Qualifications
I understand that there are many other factors that influence the answer. A general intuition would be best, but, if added complexity is unavoidable, assume:
- Serial solver
- Direct solver
- $A,\tilde A, \tilde B$ are dense
- The $m$ eigen-pairs with lowest eigenvalues are desired, where $n / m \approx 10$
- $A$ and $\tilde A, \tilde B$ model the same system, so the relative differences in the $m$ eigen-pairs are small
- $\tilde B = I + \hat{B}$ where $\text{rank}(\hat{B}) = p$, and $p \approx 2 m = n/5$
Answers for a different set of qualifications might be useful to other readers as well.