Let x_1, x_2, x_3, x_4, x_5 be a system of orthonormal vectors in R^10. Consider the matrix A = x_1x_1^ + + x_5x_5^. Which of the following statements is/are correct?

Singular values of A are also its eigenvalues

Singular values of A are either 0 or 1

Singular values of A are also its eigenvalues

Singular values of A are either 0 or 1

GATE DA 2025 Q50: Sum of five orthonormal outer products in ℝ¹⁰

Linear Algebra
2 marks
MSQ
No negative marking

Let $x_1, x_2, x_3, x_4, x_5$ be a system of orthonormal vectors in $\mathbb{R}^{10}$ . Consider the matrix $A = x_1x_1^{\top} + \ldots + x_5x_5^{\top}$ . Which of the following statements is/are correct?

Try it: multiple select: one or more correct options

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Answer: A, B (official GATE 2025 final answer key)

Concept used: $A$ is the orthogonal projection onto a 5-dimensional subspace.

First step: Check $Ax_i = x_i$ for each $x_i$ and $Av = 0$ for $v$ orthogonal to all of them. So $A$ is symmetric with eigenvalues 1 (five times) and 0 (five times).

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Source: official GATE 2025 DA question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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GATE DA 2025 Q50: Sum of five orthonormal outer products in ℝ¹⁰

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