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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 that 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 definiteness of a matrix and its determinant?

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    $\begingroup$ Note that, say, $x^2+xy+y^2$ is positive definite despite the $xy$ term. $\endgroup$
    – Gerry Myerson
    Commented Jun 11, 2012 at 13:34
  • $\begingroup$ i know , but my situation here is not simple , and i have three variables. $\endgroup$
    – the hunter
    Commented Jun 11, 2012 at 13:42
  • $\begingroup$ 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. $\endgroup$
    – user856
    Commented Jun 11, 2012 at 13:52
  • $\begingroup$ See math.stackexchange.com/questions/58010/… $\endgroup$
    – David E Speyer
    Commented Jun 11, 2012 at 18:07
  • $\begingroup$ @DavidSpeyer exactly what i want.. thanks $\endgroup$
    – the hunter
    Commented Jun 11, 2012 at 23:17

3 Answers 3

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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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  • $\begingroup$ This method is very difficult for high order matrices. $\endgroup$
    – ayr
    Commented Feb 1, 2022 at 5:06
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    $\begingroup$ @dtn almost everything is difficult for high order matrices… $\endgroup$
    – rschwieb
    Commented Feb 1, 2022 at 6:07
  • $\begingroup$ Yes, you are right, but there are so many tasks with high-order matrices... $\endgroup$
    – ayr
    Commented Feb 1, 2022 at 6:08
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Not guarenteed to yield a conclusive result but Gershgorin's Theorem may be used. See http://en.m.wikipedia.org/wiki/Gershgorin_circle_theorem

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Just do calculation of the term X^TAX and then check whether the answer can be negative or not. if it can be negative then it is not positive definite or vice versa for example if answer comes to be x1^2+x2^2+x3^2 then it can never be negative as there are squared terms so in this case matrix A will be positive definite.

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