Positive semi-definite vs positive definite

Question

I am confused about the difference between positive semi-definite and positive definite.

May I understand that positive semi-definite means symmetric and $x'Ax \ge 0$, while positive definite means symmetric and $x'Ax \gt 0$?

Answer 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