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Part BCSIR NET December 2023leading-principal-minors

Leading principal minors

For , let A_a = [[2, −1, 0], [−1, 2, −1], [0, −1, a]]. Which one of the following statements is true?

  1. A.A_a is positive definite for all a < 3.
  2. B.A_a is positive definite for all a > 3.✓
  3. C.A_a is positive definite for all a ≥ −2.
  4. D.A_a is positive definite only for finitely many values of a.

Solution

Leading principal minors: 2, 3 and det A_a = 3a − 2. Positive definite iff a > 2/3, which contains all a > 3.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

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

More on this topic

From Inner Product Spaces and Forms › Quadratic forms, positive definiteness, Sylvester's law

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