Convex quadratic functions

Theorem

Let be a quadratic function given by

  1. is convex if and only if is positive semidefinite.
  2. is strictly convex if and only if is positive definite.

Proof:

  1. Since is the Hessian of , this follows immediately from a previous theorem.
  2. From the same theorem it follows that if is positive definite then is strictly convex. Now suppose is strictly convex. From 1. we know is positive semidefinite. Suppose it is not positive definite, i.e. there exists some with . Then meaning is linear in the -direction, contradicting strict convexity.