A positive definite function is, is greatest generality, one which is strictly postiive for nonzero arguments (and zero for a zero agrument).

In quantum mechanics, an operator A is positive definite iff for all states, <p|A|p> >= 0.
An non-degenerate operator of the form B+B is always positive definite because <p|B+B |p> = <q|q>, where |q> = B|p>, and as an axiom <q|q> >= 0.

A Positive Definite Matrix can be recognized in a variety of ways:

  1. The pivots (without row exchanges) of the matrix are positive.
  2. xT*A*x > 0 for all nonzero vectors x
  3. All eigenvalues of A satisfy lambda > 0
  4. All upper left submatricies of Ak have postive determinants.

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