# Express 3x3 matrix as projection + shearing

1. Feb 25, 2012

### TomAlso

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 $\mathbb{R}^3$ 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

2. Feb 25, 2012

### alanlu

3. Feb 25, 2012

### 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..

4. Feb 25, 2012

### kai_sikorski

EDIT: never mind this was wrong

Last edited: Feb 25, 2012