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Part CCSIR NET June 2023sylvester-signature

Sylvester signature

Consider the following quadratic forms over R:(a)6X2−13\mathbb{R}: (a) 6X^{2} - 13XY +6Y2,(b)X2−+ 6Y^{2}, (b) X^{2} - XY +2Y2,(c)X2−+ 2Y^{2}, (c) X^{2} - XY −2Y2- 2Y^{2}. Which of the following statements are true?

  1. A.Quadratic forms (a) and (b) are equivalent.
  2. B.Quadratic forms (a) and (c) are equivalent.✓ Correct
  3. C.Quadratic form (b) is positive definite.✓ Correct
  4. D.Quadratic form (c) is positive definite.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests invariants don't determine.

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50 are analysed free — try those first.

The trap it tests

Invariants don't determine

Two objects sharing an invariant were treated as the same object.

Drill statements like this

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

More on this topic

The chapter behind this: Quadratic forms, signature and definiteness — free to read

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

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