Positive semi-definite vs positive definite 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 $x'Ax \gt 0$?
 A: 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. 

A: 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
A: A symmetric matrix A is said to be positive definite if for for all non zero X  $X^tAX>0$ and it said be positive semidefinite if their exist some nonzero X such that $X^tAX>=0$.
