Express 3x3 matrix as projection + shearing

  • Thread starter TomAlso
  • Start date
  • #1
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Homework Statement



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
 

Answers and Replies

  • #3
kai_sikorski
Gold Member
162
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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..
 
  • #4
kai_sikorski
Gold Member
162
0
EDIT: never mind this was wrong
 
Last edited:

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