Imagine I have a vector equation ($f:\mathbb{C}^N \mapsto \mathbb{C}$), for instance in the form: $$f(\mathbf{w}) = \frac{\mathbf{w}^H \mathbf{A} \mathbf{w}} {\mathbf{w}^H \mathbf{B} \mathbf{w}}$$
where $\mathbf{w} \in \mathbb{C}^N$ such that $\mathbf{w}^H\mathbf{w} = 1$ and $\mathbf{A},\mathbf{B} \in \mathbb{C}^{N \times N}$ with $\mathbf{B}$ positive definite. I want to define $f(\mathbf{w})$ through the matrix $\mathbf{F}$ such that: $$f(\mathbf{w}) = \mathbf{w}^H \mathbf{F} \mathbf{w}$$
In this case, $f(\mathbf{w})$ may be expressed according to the previous definition with $\mathbf{F} = \mathbf{B}^{-1/2} \mathbf{A} \mathbf{B}^{-1/2}$. Note that the eigenvalues $\lambda_i$ of $\mathbf{F}$ are the same than those of $\mathbf{F}^\prime = \mathbf{B}^{-1} \mathbf{A}$ but the eigenvectors are different, say $\mathbf{u}_i$ and $\mathbf{v}_i$, respectively. Here I am a bit confused since $f(\mathbf{w})$ needs to be defined with the $\mathbf{F}$ matrix but are the eigenvectors of $\mathbf{F}^\prime$ those that produce the values given by $\lambda_i$. That is:
$$\lambda_i = \frac{\mathbf{v}_i^H \mathbf{A} \mathbf{v}_i} {\mathbf{v}_i^H \mathbf{B} \mathbf{v}_i} \neq \frac{\mathbf{u}_i^H \mathbf{A} \mathbf{u}_i} {\mathbf{u}_i^H \mathbf{B} \mathbf{u}_i}$$
My question is how these two matrices $\mathbf{F}$ and $\mathbf{F}^\prime$ are related (more specifically their eigenvectors) in the context of the definition of the function $f(\mathbf{w})$ according to their numerical range.