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Math Help - Help with critical points

  1. #1
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    Help with critical points

    Assuming that A is a real symmetric n x n matrix, and

    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



    I feel like that's not the right answer, though.
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  2. #2
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    Re: Help with critical points

    Quote Originally Posted by tubetess123 View Post
    Assuming that A is a real symmetric n x n matrix, and

    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



    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
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  3. #3
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    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.
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  4. #4
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    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!
    Last edited by tubetess123; December 7th 2011 at 11:47 AM.
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  5. #5
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    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}.
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  6. #6
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    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.
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  7. #7
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    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!
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  8. #8
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    Re: Help with critical points

    Quote Originally Posted by tubetess123 View Post
    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
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  9. #9
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    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.
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  10. #10
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    Re: Help with critical points

    Quote Originally Posted by tubetess123 View Post
    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
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  11. #11
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    Re: Help with critical points

    Quote Originally Posted by tubetess123 View Post
    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
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  12. #12
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    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.
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  13. #13
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    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.
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  14. #14
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    Re: Help with critical points

    Quote Originally Posted by tubetess123 View Post
    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}
    Last edited by CaptainBlack; December 8th 2011 at 04:24 AM.
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  15. #15
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    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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