Express 3x3 matrix as projection + shearing

[TomAlso]

Homework Statement

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 $\mathbb{R}^3$ onto itself. Describe $A$ 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 $A$ as a product $A = BC$ where $B$ and $C$ are 3-by-3 matrices, $B$ representing a projection onto a plane and $C$ representing a shearing of such plane.

Since $B$ is a projection it must be that $B = B^2$ 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 $C$. How do we proceed after that?

Any help is greatly appreciated :D

 

[alanlu]

http://en.wikipedia.org/wiki/Shear_mapping

 

[kai_sikorski]

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..

 

[kai_sikorski]

EDIT: never mind this was wrong