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i want to know how to check if a matrix M is positive definite ,assume that M is 3x3 real numbers matrix

i think one way is to put the matrix in a quadratic form $X^TMX$ , where X is a vector $X^T=[x_1 x_2 x_3]$ , my question is if i found that $X^TMX = ax_1^2 + bx_1*x_2+ ........$ can i say the matrix M is not positive definite because the term $bx_1*x_2$ can be negative or i have to try to put the value of $X^TMX$ in the form of sum of squares e.g.,$()^2+()^2+.....$ and then decide?

and what is the relation between the positive definitness of a matrix and its determinant?

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Note that, say, $x^2+xy+y^2$ is positive definite despite the $xy$ term. –  Gerry Myerson Jun 11 '12 at 13:34
    
i know , but my situation here is not simple , and i have three variables. –  the hunter Jun 11 '12 at 13:42
    
Gerry's point is that no, you can't say that the matrix is not positive definite just because you found an $x_1x_2$ term. –  Rahul Jun 11 '12 at 13:52
    
    
@DavidSpeyer exactly what i want.. thanks –  the hunter Jun 11 '12 at 23:17

3 Answers 3

up vote 1 down vote accepted

I don't think there is a nice answer for matrices in general. Most often we care about positive definite matrices for Hermitian matrices, so a lot is known in this case.

The one I always have in mind is that a Hermitian matrix is positive definite iff its eigenvalues are all positive.

Glancing at the wiki article on this alerted me to something I had not known, Sylvester's criterion which says that you can use determinants to test (a Hermitian matrix) for positive definiteness by checking to see if all the square submatrices whose upper left corner is the $(1,1)$ entry have positive determinant.

Sorry if this is repeating things you already know, but it's the most useful information I can provide. Good luck!

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thanks for your answer, and thanks for saying "Good Luck" –  the hunter Jun 11 '12 at 23:19

Not guarenteed to yield a conclusive result by Gershgorin's Theorem may be used. See http://en.m.wikipedia.org/wiki/Gershgorin_circle_theorem

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The easiest way how to check the property is to perform a Cholesky decomposition on the matrix and concluding that the matrix is not positive definite provided that you obtain a negative pivot at some step. If you don't then the matrix is obviously s.p.d.

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