• Support PF! Buy your school textbooks, materials and every day products Here!

Hessian matrix question.

  • #1

Homework Statement



Given a function f: R^2 -> R of class C^3 with a critical point c.

Why CANNOT the hessian matrix of f at point c be given by:

1 -2
2 3


Homework Equations





The Attempt at a Solution



So first i want to clarify this.

When it says f: R^2 -> R, that means the function is of two variables (x and y)?

And when it says class C^3 that means the third derivative of the function exists and is continuous. So would a function be x^3 or x^4? the third derivative would be 24x for x^4 and is continuous. The third derivative of x^3 would be 6.

I'm not sure about the answer..
 

Answers and Replies

  • #2
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,728

Homework Statement



Given a function f: R^2 -> R of class C^3 with a critical point c.

Why CANNOT the hessian matrix of f at point c be given by:

1 -2
2 3


Homework Equations





The Attempt at a Solution



So first i want to clarify this.

When it says f: R^2 -> R, that means the function is of two variables (x and y)?

And when it says class C^3 that means the third derivative of the function exists and is continuous. So would a function be x^3 or x^4? the third derivative would be 24x for x^4 and is continuous. The third derivative of x^3 would be 6.

I'm not sure about the answer..
IF your matrix A above was a Hessian, what would the number a(2,2) = -2 represent? What would the number a(2,1) = +2 represent?

RGV
 
  • #3
Ah ok. I think i got it. In the hessian which is given by

fxx fxy

fyx fyy

fxy is not equal to fyx which should be the case for mixed partials?
 
  • #4
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,728
Ah ok. I think i got it. In the hessian which is given by

fxx fxy

fyx fyy

fxy is not equal to fyx which should be the case for mixed partials?
Yes, exactly.

RGV
 
  • #5
3
0
I have a question considering the applicability of Hessian matrix.

So, Can I use Hessian to prove that x^y > y^x whenever y > x >= e.

At first I start by multiplying by ln() => y*ln(x) > x*ln(y)

Is it enough, if I take g(x,y) such that g(x,y) = y*ln(x) - x*ln(y) > 0 and show det(H(g)) < 0 whenever y > x >= e?

My purpose with this is to show that there are no real local or global critical points in g(x,y) when y > x >= e, and conclude that x^y - y^x diverges. I am not sure if I can use Hessian to draw that kind of conclusion.
 
  • #6
lanedance
Homework Helper
3,304
2
Hi Viliperi, welcome to PF, please start a new thread if you have a question as opposed to resurrecting an old one. You're more likely to get an anser that way as well - thanks
 

Related Threads on Hessian matrix question.

  • Last Post
Replies
4
Views
3K
  • Last Post
Replies
3
Views
12K
Replies
7
Views
2K
Replies
0
Views
2K
Replies
1
Views
12K
Replies
3
Views
6K
Replies
4
Views
2K
Replies
1
Views
8K
  • Last Post
Replies
6
Views
5K
  • Last Post
Replies
1
Views
3K
Top