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Express 3x3 matrix as projection + shearing

  1. Feb 25, 2012 #1
    1. The problem statement, all variables and given/known data

    Think of the following matrix

    [itex]
    A =
    \left( \begin{array}{ccc}
    a & b & c \\
    d & e & f \\
    g & h & i \end{array} \right)
    [/itex]

    as a transformatiom of [itex]\mathbb{R}^3[/itex] onto itself. Describe [itex]A[/itex] as a projection onto a plane followed by a shearing motion of the plane.

    2. The attempt at a solution

    So the problem basically asks to rewrite [itex]A[/itex] as a product [itex]A = BC[/itex] where [itex]B[/itex] and [itex]C[/itex] are 3-by-3 matrices, [itex]B[/itex] representing a projection onto a plane and [itex]C[/itex] representing a shearing of such plane.

    Since [itex]B[/itex] is a projection it must be that [itex]B = B^2[/itex] and that is pretty much all I know. I can't seem to find precise definition of shearing as a transformation. What can we say about [itex]C[/itex]. How do we proceed after that?

    Any help is greatly appreciated :D
     
  2. jcsd
  3. Feb 25, 2012 #2
  4. Feb 25, 2012 #3

    kai_sikorski

    User Avatar
    Gold Member

    Im confused about how this could be true for a non-singular matrix, since all vectors in R3 seem to get mapped to 1 plane..
     
  5. Feb 25, 2012 #4

    kai_sikorski

    User Avatar
    Gold Member

    EDIT: never mind this was wrong
     
    Last edited: Feb 25, 2012
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