Help with critical points

Assuming that A is a real symmetric n x n matrix, and http://latex.codecogs.com/png.latex?... e^{||x||^{2}}

prove that f has a critical point at x if and only if Ax = ?.

I need to find out what Ax must equal for a critical point, and I just can't come up with anything.

I've tried taking the gradient and such, but I always end up with

http://latex.codecogs.com/png.latex?Ax = -(Ax \cdot x)x

I feel like that's not the right answer, though.

Re: Help with critical points

Quote:

Originally Posted by

**tubetess123**

Write , now write in tems of the components, ...

When you have found the gradient set it to zero, and simplify and then translate back in to vector/matrix form

CB

Re: Help with critical points

Can you please explain why multiplying by a unit vector and writing in summation notatioin works? I don't understand how this makes a difference.

Re: Help with critical points

I don't understand how I get the gradient now. Doesn't setting x as that make it a scalar? Doesn't it need to be a vector? Can you please help me? I really have no clue.

Thanks!

Re: Help with critical points

No, that doesn't make x a scalar. is just the vector written in component form. In three dimensions, it would be .

Re: Help with critical points

But how then, do I know what to do with A when I take the gradient? I learned that that gradient of is 2Ax. But then I can't figure out what Ax must be in order for there to be a critical point. I don't know how Writing the vector in component form changes anything either. I'll keep trying though.

Re: Help with critical points

I got down to . So I don't know how to determine what value of Ax will make that equation equal to 0. Also, to translate it back to regular form, is

correct?

Thanks!

Re: Help with critical points

Quote:

Originally Posted by

**tubetess123** Can you please explain why multiplying by a unit vector and writing in summation notatioin works? I don't understand how this makes a difference.

Because it turns your function into:

CB

Re: Help with critical points

I don't get what is. Can you please explain where that comes from? I also don't get what I'd do to take the gradient of that equation. Can you please explain what I need to do to take the gradient? I guess part of my problem is I don't understand what the A component of the sum is.

Thanks! Sorry for all the lack of understanding. I'm doing my best.

Re: Help with critical points

Re: Help with critical points

Quote:

Originally Posted by

**tubetess123** I also don't get what I'd do to take the gradient of that equation. Can you please explain what I need to do to take the gradient? I guess part of my problem is I don't understand what the A component of the sum is.

Thanks! Sorry for all the lack of understanding. I'm doing my best.

What definition of Gradient are you working with?

CB

Re: Help with critical points

I'm don't know what you mean by definition of gradient. I just do it as if I'm differentiating, but I call it the gradient.

Re: Help with critical points

At this point I got it down to

.

But I don't know how to simplifiy it past that point. This is where I had gotten it to when not in component form I think. But now I just don't know where to go.

I'm so confused. Thank you for all your help so far. It's so greatly appreciated.

Re: Help with critical points

Quote:

Originally Posted by

**tubetess123** At this point I got it down to

.

But I don't know how to simplifiy it past that point. This is where I had gotten it to when not in component form I think. But now I just don't know where to go.

I'm so confused. Thank you for all your help so far. It's so greatly appreciated.

Well if that is right, what you have is:

Re: Help with critical points

Is that right? I have no idea. And if it is, I still have no idea what Ax has to be for that equation to be 0.