1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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

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

    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


    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


    User Avatar
    Gold Member

    EDIT: never mind this was wrong
    Last edited: Feb 25, 2012
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook