I have a sqaure that is being transformed by the following matrix –
\begin{pmatrix} -0.2 & -0.6\\
-0.6 & 0.7
\end{pmatrix}
The original sqaure has coordinates (4,2), (2,6), (6, 8) and (8,4)
I have found the coordinates of the image as (-2, -1), (-4,3), (-6,2) and (-4, -2)
As the image has changed to a rectangle it cant be a shear – but how do I describe this transformation as the sqaure has moved and is half the width.
Also how can I find which lines are invariant under this matrix?
Many thanks
Best Answer
If I take a point $(x,y)$, the transformation gives us
$$\begin{bmatrix}-0.2 & -0.6 \\-0.6 & 0.7\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}-0.2x -0.6y \\ -0.6x + 0.7y\end{bmatrix}$$
Now for invariant points, we have
$$-0.2x -0.6y = x \implies 2x + y = 0$$
$$-0.6x +0.7y = y \implies 2x +y = 0$$
Hence the line $2x+y = 0$ is the invariant in this transformation
If you see the above transformation, it seems like a reflection about the line, but instead of being equidistant from the mirror, the image is half the distance of the object from the mirror
Try it out on desmos and see for yourself how this behaves