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Composition of Vector Functions in Mathematica

  1. Jul 26, 2012 #1
    I want to define something like:
    [tex]A\left(\begin{matrix} x \\ y\end{matrix}\right) = \left(\begin{matrix} 1 & -1 \\ -1 & 1 \end{matrix}\right)\left(\begin{matrix} x \\ y\end{matrix}\right)[/tex]
    [tex]B\left(\begin{matrix} x \\ y\end{matrix}\right) = \left(\begin{matrix} 0 & 1 \\ 2 & -1 \end{matrix}\right)\left(\begin{matrix} x \\ y\end{matrix}\right)+\left(\begin{matrix} 1 \\ 1\end{matrix}\right)[/tex]
    And then I want to be able to evaluate compositions such as [itex]A \circ B \circ A\left(\begin{matrix} x \\ y\end{matrix}\right)[/itex] quickly and easily.

    Currently I'm using this syntax:
    A[x_,y_] = {{1,-1},{-1,1}}.{{x},{y}}

    However, when I define such a function, the output is a column vector, not a list, and I can't input a column vector into the next function. How do I do ths?
     
  2. jcsd
  3. Jul 26, 2012 #2

    phyzguy

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    Science Advisor

    If I understand your syntax correctly, you have too many brackets in the column vector definition. A column vector should be just a simple list, and a matrix should be a nested list. Instead of:
    A[x_,y_] = {{1,-1},{-1,1}}.{{x},{y}}

    Just define:

    A = {{1,-1},{-1,1}}
    B = {{0,1},{2,-1}}
    X = {x,y}

    Then you can simply compose them with the dot product, such as:

    (A.(B.A)).X
     
  4. Jul 26, 2012 #3
    I can't do that, because B has a translation vector
     
  5. Jul 26, 2012 #4
    Is this what you are looking for?

    In[1]:= A[{x_,y_}]:={{1,-1},{-1,1}}.{x,y};
    B[{x_,y_}]:={{0,1},{2,-1}}.{x,y}+{1,1};
    Composition[A,B,A][{x,y}]

    Out[3]= {-2 x-2 (x-y)+2 y,2 x+2 (x-y)-2 y}
     
  6. Jul 26, 2012 #5
    I'll have to try it out when i get back tonight, but that looks very promising, thanks :)
     
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