Let $A, B \in M(n, \mathbb{R})$ be $n \times n$ symmetric square matrices with real values and suppose $B$ is invertible. We can define three bilinear forms
- $(u,v) \mapsto u^TAv$
- $(u,v) \mapsto u^TBv$
- $(u,v) \mapsto u^T(AB^{-1})v$
What is the relation between the two scalar quantities $\frac{v^TAv}{v^TBv}$ and $v^T(AB^{-1})v$ as $v$ varies in the unit ball $\{v \in \mathbb{R}^n : ||v|| = 1\}$?
The question is related to how the objective function of Linear Discriminant Analysis is formulated. In that case, $A$ is the between-class covariance matrix and $B$ is the within-class covariance matrix. In that context I read that there should be a direct proportionality between the two.