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October 21st, 2016, 05:15 PM  #1 
Member Joined: Oct 2016 From: Sverige Posts: 32 Thanks: 2  Can someone explain what Q(h,k) has to do with det(Hessian)?
Is it the same thing? Because I'm taught that in order to find out if a point on a plane in 3D space is a saddlepoint, local max or local min, then I'll have to look at what the determinant of the Hessian matrix is. But in all the answers it doesn't mention Hessian, it only says like: which is positive definite quadratic form, thus the point is a local min. From where are they getting these formulas? What is this Q(h,k)? Last edited by skipjack; October 21st, 2016 at 08:02 PM. 
October 21st, 2016, 08:10 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,835 Thanks: 2162 
They probably chose that letter because it's the first letter of "quadratic". The quadratic in your example has its minimum at the origin.

October 22nd, 2016, 07:01 PM  #3 
Member Joined: Oct 2016 From: Sverige Posts: 32 Thanks: 2 
But the whole equation is nothing similar to the result of a hessian determinant. What kind of stuff are they doing here? 
October 23rd, 2016, 03:00 AM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
It would help if you told us what the function was that gave that result!

October 23rd, 2016, 07:41 AM  #5 
Senior Member Joined: Sep 2016 From: USA Posts: 635 Thanks: 401 Math Focus: Dynamical systems, analytic function theory, numerics 
If $A$ is a positive definite matrix, then the linear functional defined by $v \mapsto v^TAv$ is nonnegative for any vector $v$. In your case, $A$ is a 2by2 matrix which is the Hessian for some scalar function and the above is testing that it is positive definite by evaluating the linear functional on the vector $v = (h,k)$.


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