The Journal of Online Mathematics and Its Applications, Volume 8 (2008)

The Most Marvelous Theorem in Mathematics, Dan Kalman

As a general matter, any 2 by 2 matrix can be used to define a transformation, and every such transformation can be understood geometrically. There are two specific examples that we shall use. The first has a matrix in one of two forms,

or .

These transformations scale a figure horizontally or vertically. For example, with `r` = 1/2 in the first matrix, we obtain the transformation

which moves each point (`x`, `y`) horizontally half way to the `y` axis. Figure 1 shows a red circle transformed in this way into a blue ellipse. Notice that the blue image has the same height as the original red circle, but is only half as wide.

The second specific kind of transformation we will use is called a shear. A shear also comes in two forms, either

or .

The first is called a horizontal shear -- it leaves the `y` coordinate of each point alone, skewing the points horizontally. In Figure 2. This is illustrated with `s` = 1, transforming a red polygon into its blue image.

The effect is something like slanting the sides of a deck of cards. Applying a horizontal shear to a rectangular grid transforms all the vertical lines into slanting lines. The new grid has the same horizontal and vertical spacing between lines, but what were vertical lines become inclined lines with slope 1/`s`. The second form of the shear transformation has a similar effect, but it operates vertically instead of horizontally.

Different matrix transformations can be combined by applying them one after another. Say you have one matrix that rotates through an angle of π/4 counter-clockwise, then a second one that scales by a factor of 1/2 horizontally, and then a third that rotates back through an angle of π/4 clockwise. The combined effect of all three transformations is to scale by a factor of 1/2 along a line that is inclined at angle of π/4. All three matrices can be combined into a single matrix that embodies this overall transformation.

There is a converse to this observation. Before getting to that, we have to observe that some matrix transformations reduce the two dimensions of the plane to a one dimensional image. As a very simple example, if you scale horizontally with a factor of 0, then the point (`x`, `y`) gets transformed to (0, `y`). That means that for any starting figure, the image is confined to the `y` axis. Such a transformation is said to be *singular*. A nonsingular transformation may deform a figure, taking a circle to an ellipse for example, but a singular transformation collapses all plane figures into a single line.

The converse of the observation given above applies to nonsingular matrices. Any such matrix transformation can be decomposed into a series of three simple geometric operations. The first and last are rotations about the origin, although not necessarily through the same angle. The middle transformation has a matrix in the form

where `r` and `s` are nonzero constants. This has the effect of scaling horizontally by a factor of `r` and vertically by a factor of `s`. If either scale factor is negative, the image is also reflected through an axis. For example, with a horizontal scale factor of −2, every point gets reflected through the `y` axis, and then stretched away from the `y` axis by a factor of 2.