GATE DA 2024 — Question 22

For any binary classification dataset, let \(S_B \in \mathbb{R}^{d \times d}\) and \(S_W \in \mathbb{R}^{d \times d}\) be the between-class and within-class scatter (covariance) matrices, respectively. The Fisher linear discriminant is defined by \(u^{*} \in \mathbb{R}^{d}\), that maximizes

\(J(u) = \dfrac{u^{T} S_B u}{u^{T} S_W u}\)

If \(\lambda = J(u^{*})\), \(S_W\) is non-singular and \(S_B \neq 0\), then \((u^{*}, \lambda)\) must satisfy which ONE of the following equations?

Note: \(\mathbb{R}\) denotes the set of real numbers.