GATE DA 2025 — Question 60

Let \(D = \{x^{(1)}, \dots, x^{(n)}\}\) be a dataset of \(n\) observations where each \(x^{(i)} \in \mathbb{R}^{100}\). It is given that \(\sum_{i=1}^{n} x^{(i)} = 0\). The covariance matrix computed from \(D\) has eigenvalues \(\lambda_i = 100^{\,2-i}\), \(1 \le i \le 100\). Let \(u \in \mathbb{R}^{100}\) be the direction of maximum variance with \(u^{T} u = 1\).

The value of \(\dfrac{1}{n} \sum_{i=1}^{n} \big(u^{T} x^{(i)}\big)^{2}\) is ______ (answer in integer).