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February 29th, 2016, 10:45 AM  #1 
Newbie Joined: Feb 2016 From: san jose, ca Posts: 1 Thanks: 0 Math Focus: wide ranging as other interests dictate  verifying matrix is positive definite
Suppose I have a symmetric matrix. I need to check that this matrix is 'positive definite'. I am not a mathematician, but have a basic understanding of how to solve linear equations using gaussian elimination and backward substitution (pretty basic stuff). I have not been able to find a simple test for 'positive definite'. One source indicates that the matrix is positive definite if the matrix' determinant is positive. That seems too simple to me. It may be that any positive definite matrix has a positive determinant but not all matrices which have a positive determinant are positive definite? I'm looking for some help here. The best I've been able to find is that the matrix is positive definite if and only if all the pivot points are positive. That is an understandable test for me as it relates to the mechanisms I use to solve linear equations. Thanks for any help. ward Last edited by skipjack; March 5th, 2016 at 11:17 PM. 
March 5th, 2016, 09:45 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
The simplest characterization of "positive definite" matrix is that all its eigenvalues are positive real numbers. If all its eigenvalues are positive definite then the determinant of the matrix is positive definite but the converse is not true. If there are an even number of negative eigenvalues then the matrix will be positive definite.
Last edited by skipjack; March 5th, 2016 at 11:15 PM. 
March 5th, 2016, 11:19 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,924 Thanks: 2203  

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