How do I find eigenvalues of an hessian matrix that depends on (x,y,z)?

Question

For example:

$f(x,y,z)=1+z^2+sin(z)+x^2+|y|^3$

Hf $(x,y,z) = diag(2, 6y, 2- sin(z))$

Well, how do I calculate the eigen values? Is it going to be in function of y and z? So they won't be fixed?

If whenever I find $eig(x,y,z) > 0$ for every $(x,y,z)$, only then I can be sure it is convex?

I'm a little bit confused: my teacher says that this function has a unique global minimum, but I'm quite sure it doesn't.

Thank you all!!

score 1 · Accepted Answer · answered Nov 29 '18 at 15:02

1

A quick comment first,

$$ \frac{\partial |y|^3}{\partial y} \not= 2y $$

as a matter of fact at $y = 0$ the whole expression just break. You can follow this other post to see more details. That being said, it is not a problem if the Hessian and its eigenvalues explicitly depend on the coordinates, it just tells you that the geometry changes with location.

answered Nov 29 '18 at 15:02

caverac

19,783

actualy it is 6y, sorry my bad, but anyways the problem stands still – Lucas Tonon Nov 29 '18 at 15:05
Well, I can see that the geometry changes, but can I define the eigenvalues as functions of (x,y,z)? – Lucas Tonon Nov 29 '18 at 15:06
1

@LucasTonon Yes, absolutely. There's not a problem with that – caverac Nov 29 '18 at 15:06
it's 6ysign(y) actually cause y^3 is smooth in the origin, so it is derivable in zero and has derivative zero, so |y|^3 derived may be sign(y)3y^2 and that derived is 6y*sign(y), my bad – Lucas Tonon Nov 29 '18 at 15:11

How do I find eigenvalues of an hessian matrix that depends on (x,y,z)?

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