Positive semi-definite vs positive definite

Solution 1:

Yes. In general a matrix $A$ is called...

• positive definite if for any vector $x \neq 0$, $x' A x > 0$
• positive semi definite if $x' A x \geq 0$.
  • nonnegative definite if it is either positive definite or positive semi definite
• negative definite if $x' A x < 0$.
• negative semi definite if $x' A x \leq 0$.
  • nonpositive definite if it is either negative definite or negative semi definite

• indefinite if it is nothing of those.

  Literature: e.g. Harville (1997) Matrix Algebra From A Statisticians's Perspective Section 14.2

Solution 2:

A great source for results about positive (semi-)definite matrices is Chapter 7 in Horn, Johnson (2013) Matrix Analysis, 2nd edition. One result I found particularly interesting:

Corollary 7.1.7. A positive semidefinite matrix is positive definite if and only if it is nonsingular.