1. Help with critical points

Assuming that A is a real symmetric n x n matrix, and $f(x) = (Ax \cdot x) e^{||x||^{2}}$

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

$Ax = -(Ax \cdot x)x$

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

2. Re: Help with critical points

Originally Posted by tubetess123
Assuming that A is a real symmetric n x n matrix, and $f(x) = (Ax \cdot x) e^{||x||^{2}}$

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

$Ax = -(Ax \cdot x)x$

I feel like that's not the right answer, though.
Write $x=\sum_i x_i \hat{e}_i$, now write $f(x)$ 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

3. 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.

4. 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!

5. Re: Help with critical points

No, that doesn't make x a scalar. $\sum_i x_i\hat(e)_i$ is just the vector written in component form. In three dimensions, it would be $\vec{x}= x_1\vec{i}+ x_2\vec{j}+ x_3\vec{k}$.

6. 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 $Ax \cdot x$ 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.

7. Re: Help with critical points

I got down to $0 = Ax_i\hat{e}_i + x_i\hat{e}_i (Ax \cdot x)$. 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
$0 = Ax + x(Ax \cdot x)$ correct?

Thanks!

8. Re: Help with critical points

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:

$f({\bf{x}})=\left[ \sum_i \sum_j A_{i,j}x_i x_j\right]e^{\sum_kx_k^2}$

CB

9. Re: Help with critical points

I don't get what $A_{ij}$ 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.

10. Re: Help with critical points

Originally Posted by tubetess123
I don't get what $A_{ij}$ 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.
$A_{i,j}$ is the value in the $i$-th row and $j$-th column of $A$.

Note: because of the given condition $A_{i,j}$ is real and $A_{i,j}=A{j,i}$

CB

11. Re: Help with critical points

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

12. 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.

13. Re: Help with critical points

At this point I got it down to

$0 = \sum_j{A_{ij}x_{j}}+x_{i}\sum_{m,j}{A_{mj}x_{m}x_{ j}$.

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.

14. Re: Help with critical points

Originally Posted by tubetess123
At this point I got it down to

$0 = \sum_j{A_{ij}x_{j}}+x_{i}\sum_{m,j}{A_{mj}x_{m}x_{ j}$.

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:

${\bf{Ax}}+({\bf{x^tAx }}){\bf{x}}=\bf{0}$

$[{\bf{A}}+({\bf{x^tAx}}){\bf{I}}]{\bf{x}}=\bf{0}$

15. 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.

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