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Quadratic forms under constraints

  1. Feb 28, 2016 #1
    • Member warned about posting with no effort shown
    1. The problem statement, all variables and given/known data
    Find the minimum value of ## x_1^2+x_2^2+x_3^2## subject to the constraint:
    ## q(x_1,x_2,x_3)=7x_1^2+3x_2^2+7x_3^2+2x_1x_2+4x_2x_3=1 ##

    2. Relevant equations


    3. The attempt at a solution
    I am not really sure how to think about it. I have seen the opposite way but have not seen this type of question yet. Any guidance will be very helpful.

    Thank you.
     
  2. jcsd
  3. Feb 28, 2016 #2

    Ray Vickson

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    What do you mean by "the opposite way"?
     
  4. Feb 28, 2016 #3
    I mean something like "find the max/min of ## q(x_1,x_2,x_3)=7x_1^2+3x_2^2+7x_3^2+2x_1x_2+4x_2x_3## under the constraint ## x_1^2+x_2^2+x_3^2=1##"
    I know how to solve this type of questions. This is the opposite way(maybe "way" is not a good word here)
     
    Last edited: Feb 28, 2016
  5. Feb 28, 2016 #4

    Samy_A

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    How would you solve the "opposite way" problem? And why does that method not work in the present case?
     
  6. Feb 28, 2016 #5
    Because there is a theorem that says that at the unit sphere(## x_1^2+x_2^2+x_3^2=1##) the max/min of the equation is at the max/min of the eigenvalues of the form. I cannot use it here.
     
  7. Feb 28, 2016 #6

    Ray Vickson

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    Yes, you can.

    You can change variables to ##u_i = U_i(x_1,x_2,x_3)## for ##i = 1,2,3##, where the ##U_i## are linear functions of the ##x_j##, devised so that your ##q(x_1,x_2,x_3)## has the form ##u_1^2 + u_2^2 + u_3^2##. Then, if you reverse the transformations to get ##x_i = X_i(u_1,u_2,u_3)##, the functions ##X_i## will be linear in the ##u_j##, and so ##f(x_1,x_2,x_3) = x_1^2 + x_2^2 + x_3^2## will be quadratic in the ##u_j##. That problem will be exactly of the type you can solve already.

    However, I don't know why you would ever want to do this; it is much easier to just solve the problem directly using the Lagrange multiplier method.
     
  8. Feb 28, 2016 #7
    Thank you for the answer. I am not sure that I got it 100% but I will work on it.
    This is a question from a past exam in linear algebra. The subject of Lagrange multiplier is not covered in this course. I am not sure if I want to start talking about the course. I have a lot of bad things to say...

    By the way, now I understand the reason why I couldn't find any explanation or notes about something similar to that...

    Thank you for the help, I appreciate it!

    Thomas
     
    Last edited: Feb 28, 2016
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