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GATE 2025 · session-5
Linear AlgebraEigenvalues and Covariance MatrixmediumNAT2 marks
Let \( D = \{ x^{(1)}, x^{(2)}, \dots, x^{(n)} \} \) 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 \( \lambda_i = 100^2 - i \), for \( 1 \leq i \leq 100 \). Let \( u \in \mathbb{R}^{100} \) be the direction of maximum variance with \( u^T u = 1 \). The value of \[ \frac{1}{n} \sum_{i=1}^{n} \left( u^T x^{(i)} \right)^2 = \hspace{2cm} \quad (Answer in integer) \]
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