GATE 2025 DA – Question 60
Let $D = \{x^{(1)}, \ldots, x^{(n)}\}$ 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 $u^\top u = 1$.
The value of $\frac{1}{n}\sum_{i=1}^{n}\left(u^\top x^{(i)}\right)^2 = $ ______ (*Answer in integer*)
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Correct answer: 100
Explanation
The data have zero mean, so $\frac{1}{n}\sum (u^\top x^{(i)})^2$ is the variance of the data along $u$, which is $u^\top\Sigma u$. For the direction of maximum variance this is the largest eigenvalue, $\lambda_1 = 100^{2-1} = 100$.