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Expressing a quadratic form in canonical form using Lagrange

  1. Dec 19, 2016 #1
    • Moved from technical forums, so no template
    Problem:
    Express the quadratic form:

    q=x1x2+x1x3+x2x3

    in canonical form using Lagrange's Method/Algorithm

    Attempt:
    Not really applicable in this case due to the nature of my question

    The answer is as follows:
    Using the change of variables:
    x1=y1+y2
    x2=y1-y2
    x3=y3
    Indeed you get q'=(y1+y3)2-(y2)2-(y3)2

    I can see that when you substitute it in it does indeed reduce it to canonical form however my question is as follows:
    How am I meant to arrive at the above change of variables? I have seen a few cases of completing the square, but to my knowledge we can't use that here. EDIT: We can

    Any help is greatly appreciated!!

    EDIT:
    I think I've been able to deduce it....
    x1x2+x1x3+x2x3=(x3)2+x3(x2+x1)-(x3)2 and from there I completed the square, the only thing I'm still a bit uncertain on is that I worked this out the other way i.e. I worked out y in terms of x and not x in terms of y....
     
    Last edited: Dec 19, 2016
  2. jcsd
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