Let u = bmatrix 1 \\ 2 \\ 3 \\ 4 \\ 5 bmatrix, and let _1, _2, _3, _4, _5 be the singular values of the matrix M = uu^T (where u^T is the transpose of u). The value of _i=1^5 _i is ______.

GATE DA 2024 Q61: Sum of singular values of uuᵀ

Linear Algebra
2 marks
NAT
No negative marking

Let $\mathbf{u} = \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \end{bmatrix}$ , and let $\sigma_1, \sigma_2, \sigma_3, \sigma_4, \sigma_5$ be the singular values of the matrix $\mathbf{M} = \mathbf{u}\mathbf{u}^T$ (where $\mathbf{u}^T$ is the transpose of $\mathbf{u}$ ). The value of $\sum_{i=1}^{5} \sigma_i$ is ______.

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Answer: 55 (official GATE 2024 final answer key)

Concept used: $\mathbf{u}\mathbf{u}^T$ has rank 1; its only non-zero singular value is $\|\mathbf{u}\|^2$ .

First step: $\mathbf{u}\mathbf{u}^T \mathbf{u} = \|\mathbf{u}\|^2 \mathbf{u}$ , and $\|\mathbf{u}\|^2 = 1 + 4 + 9 + 16 + 25$ .

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Source: official GATE 2024 DA question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key

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GATE DA 2024 Q61: Sum of singular values of uuᵀ

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