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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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    $\begingroup$ yes pretty much. $\endgroup$ – jacob smith Apr 8 '16 at 18:36
  • $\begingroup$ I fixed some formatting issues, but you could improve the Question by opening with a mention that you are asking about properties of matrices. In any case I added that as a tag. $\endgroup$ – hardmath Apr 8 '16 at 18:41
  • $\begingroup$ The inequality for positive definite is often given as $x^TAx\ge a\gt0$, giving a positive lower bound. $\endgroup$ – robjohn Apr 8 '16 at 18:42
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    $\begingroup$ Also: for positive definite, that condition only applies when $x \ne 0$. $\endgroup$ – John Hughes Apr 8 '16 at 18:58
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    $\begingroup$ @WillJagy: ah, good point. Thanks for the correction. $\endgroup$ – robjohn Apr 8 '16 at 19:31
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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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    $\begingroup$ Aren't positive semidefinite matrices already a superset of positive definite matrices? So nonnegative definite and positive semidefinite are the same. $\endgroup$ – Rahul Jun 1 '16 at 14:57
  • $\begingroup$ Yes, they are the same; but as you can read the expression from time to time (in the mentioned literatur or in this question), i stated it. $\endgroup$ – Qaswed Jun 1 '16 at 15:16
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    $\begingroup$ That's true, but it would be clearer if you combined the two definitions instead of having them in separate bullet points as though they were different. So something like "positive semidefinite or nonnegative definite if $x^T Ax\ge 0$", and similarly for negative semidefinite / nonpositive definite. $\endgroup$ – Rahul Jun 3 '16 at 7:34
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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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  • $\begingroup$ Welcome to MSE. Your answer adds nothing new to the already existing answers. $\endgroup$ – José Carlos Santos Dec 11 '18 at 14:01
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    $\begingroup$ Sorry, but I didn't seen the above existing answer. $\endgroup$ – MANI SHANKAR PANDEY Dec 11 '18 at 14:09

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