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Wierd sum of squares compilation to a single vector norm

  1. Apr 17, 2012 #1
    I've encountered a function like this:
    S(x) = [M(x) - F(x)] ^2 + || G(x) || ^ 2


    X being a 3*1 vector
    M and F: vector----->scalar
    G: vector------->vector and || G || meaning its norm

    To change S(x) into a single square, authors have described it like this:

    S(x) = || A + B || ^ 2 where A=(M(x) - F(x) , 0) and B=(0 , G(x))

    I don't understand how two vectors could be added actually resulting a vector with the first eigenvalue being scalar and the second eigenvalue a 3*1 vector itself?
    i.e. what is the "nature" of A + B? a vector?
     
  2. jcsd
  3. Apr 17, 2012 #2

    Office_Shredder

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    Why are you describing the entries of the vector as eigenvalues?

    All that they are doing is this. Suppose that
    [tex] G(x)=(G_1(x),...,G_m(x))[/tex]
    where each Gi takes a vector to a scalar. Then
    [tex] S(x) = (M(x)-F(x))^2+G_1(x)^2+...+G_m(x)^2[/tex]
    We are adding the squares of a bunch of numbers together. This is the same as taking the norm of a vector which has those numbers as its entries
    [tex] S(x) = ||\left( M(x)-F(x),G_1(x),....,G_m(x) \right) ||^2 [/tex]

    This vector whose norm we are taking is equal to A+B in your post
     
  4. Apr 17, 2012 #3
    Re: weird sum of squares compilation to a single vector norm

    Thanks this sheds some light. Forgive my rudimentary mistakes since I am actually a doctor!

    Well, the problem was more complicated than I described:

    A(x) = (F(x) - M(x) + ∂M/∂x * x )2 taking 3*1 vector x to scalar
    B(x) = || K.x ||2 k:3*3 matrix

    I am confused how can (A + B) be represented as :

    || (F(x) - M(x) , 0) + (∂M/∂x , K).x ||2

    Especially, what would be the dimension of the matrix (∂M/∂x , K) ? (discrepancy between the first 1 * 3 entry and second 3 * 3?)
     
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