Question

For any binary classification dataset, let $S_B\in \R^{d\times d}$ and $S_W\in \R^{d\times d}$ be the between-class and within-class scatter (covariance) matrices, respectively. The Fisher linear discriminant is defined by $u^* \in \R ^d$, that maximizes
$J(u)=\frac{u^TS_Bu}{u^TS_Wu}$
If λ = J(u*), SW is non-singular and SB ≠ 0, then (u*, λ) must satisfy which ONE of the following equations ?
Note : R denotes the set of real numbers.

• A. $S_W^{-1}S_Bu^*=\lambda u^*$
• B. $S_Wu^*=\lambda S_Bu^*$
• C. $S_BS_Wu^*=\lambda u^*$
• D. $u^{*T}u^*=\lambda^2$

Answer 1

Answer: (A)

$S_W^{-1}S_Bu^*=\lambda u^*$

Answer 2

The correct option is (A) : $S_W^{-1}S_Bu^*=\lambda u^*$.