GATE 2024 DA – Question 61
Let $u = [1, 2, 3, 4, 5]^T$, and let $\sigma_1, \sigma_2, \sigma_3, \sigma_4, \sigma_5$ be the singular values of the matrix $M = uu^T$ (where $u^T$ is the transpose of $u$). The value of $\sum_{i=1}^{5}\sigma_i$ is ______.
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Correct answer: 55
Explanation
The matrix $M = uu^T$ has rank 1, so it has only one non-zero singular value. It is symmetric and positive semi-definite with the single non-zero eigenvalue $u^Tu = 1 + 4 + 9 + 16 + 25 = 55$, so $\sigma_1 = 55$ and the others are 0. The sum is 55.