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GATE 2025 DA – Question 50

Linear Algebra · Special matrices: projection, orthogonal and idempotent, quadratic forms · 2 marks · Multiple select

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?

  1. Singular values of $A$ are also its eigenvalues
  2. Singular values of $A$ are either 0 or 1
  3. Determinant of $A$ is 1
  4. $A$ is invertible

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Show answer and explanation

Correct answer: (A) Singular values of $A$ are also its eigenvalues; (B) Singular values of $A$ are either 0 or 1

Explanation

$A$ is the orthogonal projection onto the 5-dimensional space spanned by the $x_i$. It is symmetric, with the eigenvalue 1 five times (on that space) and 0 five times (on its complement). For a symmetric positive semi-definite matrix the singular values equal the eigenvalues, so A and B are true. The determinant is 0 and the matrix is not invertible, so C and D are false.