Q: Which one of the following statements is correct in the context of quadratic forms V = xᵀAx, where x = [x₁, x₂ … xₙ]ᵀ?
(a)V < 0 for all vectors x except x = 0, if and only if all the eigenvalues of A are positive
(b)V ≤ 0 for all vectors x and V = 0 for at least one vector x ≠ 0, if and only if all the eigenvalues of A are non-negative and at least one of the eigenvalues is zero
(c)V is negative-definite if −V is positive-definite, with a corresponding condition on the eigenvalues of A
(d)V is negative-semi-definite if −V is positive-semi-definite, with a corresponding condition on the eigenvalues of −A
Correct Answer: (c)
The correct answer is (c): V is negative-definite if −V is positive-definite, with a corresponding condition on the eigenvalues of A. Subject: International Relations | Sub-topic: International Groupings Note: This answer was determined through AI-based research. Verify against the official answer key when available.