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Part CCSIR NET June 2023sylvester-signature

Sylvester signature

Consider the following quadratic forms over XY XY XY . 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.✓
  3. C.Quadratic form (b) is positive definite.✓
  4. D.Quadratic form (c) is positive definite.

Solution

Discriminants ac: (a) 169 − 144 > 0 indefinite; (b) 1 − 8 < 0 with a > 0 ⇒ positive definite; (c) 1 + 8 > 0 indefinite. Over , non-degenerate binary forms are equivalent iff they have the same signature, so (a) ~ (c).

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

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

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