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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 $x'Ax \gt 0$?

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3 Answers

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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

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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.

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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 $X^tAX>=0$.

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