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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2$\begingroup$ yes pretty much. $\endgroup$– jacob smithCommented Apr 8, 2016 at 18:36
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$\begingroup$ 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. $\endgroup$– hardmathCommented Apr 8, 2016 at 18:41
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1$\begingroup$ Also: for positive definite, that condition only applies when $x \ne 0$. $\endgroup$– John HughesCommented Apr 8, 2016 at 18:58
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1$\begingroup$ @robjohn the OP did not specify that $x$ was a unit vector (and does not seem to be thinking in such a manner), so it should be $x^T A x \geq a |x|^2$ with some fixed $a > 0.$ $\endgroup$– Will JagyCommented Apr 8, 2016 at 19:08
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2$\begingroup$ @WillJagy: ah, good point. Thanks for the correction. $\endgroup$– robjohn ♦Commented Apr 8, 2016 at 19:31
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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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6$\begingroup$ Aren't positive semidefinite matrices already a superset of positive definite matrices? So nonnegative definite and positive semidefinite are the same. $\endgroup$– user856Commented Jun 1, 2016 at 14:57
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$\begingroup$ 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. $\endgroup$– QaswedCommented Jun 1, 2016 at 15:16
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5$\begingroup$ 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. $\endgroup$– user856Commented Jun 3, 2016 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.
answered Feb 22, 2017 at 13:07
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1$\begingroup$ I do not think "definite" on its own has a definition. It just means "positive semidefinite or negative semidefinite". In the quote above "positive definite" is a single expression (like "hot dog"). $\endgroup$ Commented Jul 26, 2023 at 13:44
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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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3$\begingroup$ Welcome to MSE. Your answer adds nothing new to the already existing answers. $\endgroup$ Commented Dec 11, 2018 at 14:01
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1$\begingroup$ Sorry, but I didn't seen the above existing answer. $\endgroup$– MANICommented Dec 11, 2018 at 14:09