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GATE 2024 ST – Question 48

Matrix Theory · Eigenvalues and Matrix Decompositions · 2 marks · Multiple select

For a real n×n A, which hold?

  1. If symmetric A+εI is positive semidefinite for every ε>0, then A is positive semidefinite.
  2. For odd n, A−Aᵀ is singular.
  3. A symmetric matrix with all singular values positive is positive definite.
  4. If every singular value is 1, A is orthogonal.

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Show answer and explanation

Correct answer: (A) If symmetric A+εI is positive semidefinite for every ε>0, then A is positive semidefinite.; (B) For odd n, A−Aᵀ is singular.; (D) If every singular value is 1, A is orthogonal.

Explanation

For A, vᵀAv≥−ε‖v‖² for every ε>0, so let ε↓0. An odd-dimensional skew-symmetric matrix has zero determinant, proving B. Positive singular values imply nonsingularity, not positive eigenvalues: −I refutes C. In D, AᵀA is symmetric with all eigenvalues 1 and hence equals I. **A,B,D**.

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