Exercise 12.6 - Derivation of Fisher's linear discriminant

This is but straightforward algebra:

Note that minimizing the Fisher loss is but one option in maximizing ${\mathbf{\text{𝐰}}}^{\text{T}}{\mathbf{\text{𝐒}}}_B\mathbf{\text{𝐰}}$ while minimizing ${\mathbf{\text{𝐰}}}^{\text{T}}{\mathbf{\text{𝐒}}}_W\mathbf{\text{𝐰}}$. Modern numerical toolkits offer more choices to be explored.