Pattern Recognition

$S_{\text{B}}w=\left({\overline{m}}_1-{\overline{m}}_2\right)\underset{\text{scalar}}{\underbrace{{\left({\overline{m}}_1-{\overline{m}}_2\right)}^{T}w}}\left(2.234\right)$

always has the same direction as (m1 − m2. Therefore, Equation (2.233) can be sim plfied by setting SB w = λ′ (m1− m2) for some unknown scalar λ′,

$\begin{array}{l}{S}_w^{-1}\lambda \prime \left({\overline{m}}_1-{\overline{m}}_2\right)=\lambda w\\ \iff w={\lambda }^{-1}\lambda \prime S_w^{-1}\left({\overline{m}}_1-{\overline{m}}_2\right).\left(2.235\right)\end{array}$

By virtue of the normalization condition wTSW w = 1 from Equation (2.230), the unknown factor can be cancelled:

$\begin{array}{l}w=\frac{w}{\sqrt{w^T{S}_{w}w}}=\frac{S_w^{-1}\left({\overline{m}}_1-{\overline{m}}_2\right)}{\sqrt{{\left({\overline{m}}_1-{\overline{m}}_2\right)}^T{S}_w^{-1}{T}^{}{S}_w{S}_w^{-1}\left({\overline{m}}_1-{\overline{m}}_2\right)}}\end{array}$