NETMaths
Part CCSIR NET December 2023

Quadratic forms, positive definiteness, Sylvester's law

Let A be a real diagonal matrix with characteristic polynomial . Define a bilinear form ⟨v, w⟩ = vᵗAw on . Which of the following statements are true?

  1. A.A is positive definite.
  2. B. is positive definite.
  3. C.There exists a nonzero such that ⟨v, v⟩ = 0.
  4. D.rank A = 2.

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Related counterexample: Every real symmetric matrix is positive definite if det > 0

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