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Homework Help: Prove that a function is the quadratic form associated to

  1. Feb 19, 2014 #1
    1. The problem statement, all variables and given/known data

    Let G:R2[itex]\rightarrow[/itex]R be a C2 function such that G(tx,ty)=t2G(x, y). Show that:


    3. The attempt at a solution

    G is C2, so its Taylor expansion is:

    G(x,y) = G(0,0) + [itex]\nabla[/itex]G(0,0).(x,y) + [itex]\frac{1}{2}[/itex](x,y).HG(c).(x,y)t,

    where c lies on the line segment that goes from (0,0) to (x,y).

    Using that G(tx,ty)=t2G(x,y) I get that G(0,0) and the linear term equals 0.

    The problem is that I have HG(c) in the quadratic term, but I need HG(0,0).
  2. jcsd
  3. Feb 20, 2014 #2


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    If c is not (0,0) then what do you mean by "c" and "HG(c)"? You are taking the Taylor expansion at (0, 0), are you not?
  4. Feb 20, 2014 #3
    I can't increase the degree of the Taylor polynomial because G is C2, so the second degree term is the remainder written in matrix notation.

    HG(c) is the Hessian matrix of G evaluated at c, where c lies on the segment that goes from (0,0) to (x,y).
  5. Feb 20, 2014 #4
    I solved the problem deriving two times the function f(t) = G(tx, ty). Thanks.
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