how to set Equalizer's coefficient using generalized eigenvector.
The generalized eigenvalue problem is given by

$$Bw=\lambda Cw\tag{1}$$

where $\lambda$ is the generalized eigenvalue of the matrices $B$ and $C$. Multiplying $(1)$ from the left with $w^H$ (with $^H$ denoting the Hermitian conjugate) and dividing both sides by $w^HCw$ (assuming that this term is non-zero), we obtain

$$\frac{w^HBw}{w^HCw}=J(w)=\lambda\tag{2}$$

This shows that the value of the objective function $J(w)$ (which is a generalized Rayleigh quotient) equals the generalized eigenvalue $\lambda$. Consequently, $J(w)$ is maximized by the eigenvector corresponding to the maximum eigenvalue, and it is minimized by the eigenvector corresponding to the minimum eigenvalue.