Thread: Help with critical points

Originally Posted by Hartlw

the vector x is the same in either basis. its representation by coordinates is different in different bases.
for example: u = x1e1 + x2e2 =x1'e1' + x2'e2'

Don't bother with primed coordinates. just make a note you are switching to eigenvector basis and the components are wrt that basis. The tip-off for switching to an eigenvector basis was that A is given as real and symmetric.

First solve the problem in the simple intelligible form:

f= [(ax^2+by^2]e^(x^2+y^2), x,y components of x in eigenvector system and a,b eigenvalues.

to see what happens. Then you can clean it up. I assume you can takle partials wrt x and y and set them equal to 0?

This is what I get:
let z = x^2+y^2

df/dx= axe^z + 2x(ax^2+by^2)eZ = 0
df/dy = bye^z + 2y(ax^2+by^2)e^z = 0

solve to get x=0 and llxll = 1 as the critical points which is the same in either coordinate system which we anticipated, so you don't have to worry about changing from one basis to the other.

I don't know how you got that ||x|| = 1 is a critical point. I got that either lambda or x for each i must be 0 (one or the other or both must be zero), so then Dx = 0 because Dx = the sum of lambda times x for each i. Since one of those two values is zero for all i, then that will always be zero.

I justified changing back because changing Dx back to the original basis involves matrix multiplication only, which 0 times and matrix is still 0. Thus Ax = 0 as well.

So now how do I calculate the hessian matrix of f at a critical point in "very simple terms"?

Originally Posted by tubetess123

So now how do I calcular the hessian matrix of f at a critical point in "very simple terms"?

Sorry, I don't know what a Hessian matrix is. Doesn't really interest me. I alreasdy have more than I can remember. Look up definition

Originally Posted by Hartlw

Sorry, I don't know what a Hessian matrix is. Doesn't really interest me. I alreasdy have more than I can remember. Look up definition

Curiousity got the better of me and I looked it up. Its the matrix whose elements are the second partials of f. Sounds easy enough. Do it for the simple case above and see what comes out. The partial derivatives are easy enough.