I have a problem understanding a solution to a problem. Here's the background. I am looking at the criterion function for Fisher's linear discriminant function:
$J(w) = \tfrac{w^tS_Bw}{w^tS_Ww}$
I have shown that the above function can be given by using
$S_Bw = \lambda S_Ww$
By theory, at the extremum of Fisher's linear discriminant, a small change in w must leave J(w) unchanged, that is, $J(w) = J(w + \Delta w)$. Knowing the fact that a small perturbation should leave it unchanged, I need to derive $S_Bw = \lambda S_ww$. Fortunately, I have a solution for the problem, but I can't for the life of me understand the mathematics behind it, and I'm asking for help to understand the manipulations of the matrices within the calculations. I have tried different expansions and manipulations, but I can't seem to find the same answer (and I'm starting to think there is a mistake with the solution). Here is the solution that the manual gives:
More specifically, I am having trouble to understand this exact part:
I do not understand how they first expanded to
$\tfrac{w^tS_Bw + 2\Delta w^tS_Bw}{w^tS_Ww + 2 \Delta w^tS_Ww}$
Though I'm guessing it's a Taylor Expansion to the first order, and secondly, I do not understand how they simplified it to a mere
$\tfrac{\Delta w^tS_Bw}{\Delta w^tS_Ww}$
Thanks in advance for any help you may provide.