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The inverse of a $\frac{\text{linear}}{\text{linear}} $ function

$$f(x)=\frac{ax+b}{cx+d}\implies f^{-1}(x)=\frac{dx-b}{-cx+a}, ad-bc\neq0$$

has a very similar structure to that of the inverse of a matrix (one is a multiple of the other)

$$X=\begin{bmatrix}a & b\\c & d\end{bmatrix}\implies X^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d & -b\\-c & a\end{bmatrix}, ad-bc\neq0$$

I understand these two properties, and their proofs, individually.I understand how to use matrices as representations of systems of linear equations, but am not sure that is the correct interpretation here. What would be an appropriate interpretation of the matrix, and what causes this connection?

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  • $\begingroup$ The formula for the inverse of the matrix is only true if the determinant is exactly 1. $\endgroup$
    – Andrei
    Commented Dec 11, 2021 at 4:51
  • $\begingroup$ That is because your $\Rightarrow$ is a homomorphism of the group $PSL_2(\Bbb R)$. $\endgroup$
    – markvs
    Commented Dec 11, 2021 at 5:02

1 Answer 1

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This has to do with projective geometry. Fractional linear maps natively act on projective lines. So a short excursion into projective geometry is in order.

An affine space over the reals is what you're probably familiar with: an affine line is a line, an affine plane is a plane, etc.. We get the projective variant by adding an "object at infinity", at which previously parallel lines can intersect. To the affine line we add a point at infinity, to the affine plane we add a line at infinity, to the affine space we add a plane at infinity, and so on. Consider this picture (the top half), for instance:

enter image description here

Blue is the projective plane, with the line at infinity shown to the right. We can identify each point on this plane as a projection of a point on the unit sphere (red) through the sphere's center: Draw a line through the sphere's center and see where it intersects the plane and the sphere. The two points where it intersects the sphere can be jointly identified as the corresponding point on the plane (horizontal lines intersect the plane at infinity, also pictured). In particular, we can use the coordinates of one of the two points in 3d space to give coordinates to the points on the plane.

The same works for projective lines (the lower picture). They are lines with a point at infinity, and by projecting points on the unit circle onto the line, we can give the line a coordinate system: If $(x_1,x_0)$ (the coordinate system's origin is the center of the circle) is projected to a point on the plane, then we denote that point as $[x_1,x_0]$. Note that there are multiple equivalent ways to give the same point a coordinate: The coordinates of any point on the line through which we project can be used, except the point at the center of the circle. We call these coordinates homogenous.

Homogenous coordinates with $x_0\neq 0$ describe those points on the projective line which were already part of the (affine) real number line: $[x_1,x_0]$ corresponds to the number $\frac{x_1}{x_0}$. Meanwhile, homogeneous coordinates with $x_0=0$ all denote the point at infinity (intuitively, you could think of it as $\frac{x_1}{x_0}$ still, where division by $0$ yields infinity).

Now if we apply the fractional linear map $f:x\mapsto\frac{ax+b}{cx+d}$ to a point $x$ which is given in affine coordinates as a point $[x_1,x_0]$, we get

$$f(x)=f\left(\frac{x_1}{x_0}\right)=\frac{ax_1+bx_0}{cx_1+dx_0}.$$

The last step skipped some elementary algebra. Now this new point can also be given in homogeneous coordinates: $[ax_1+bx_0,cx_1+dx_0]$. So we're mapping the point $[x_1,x_0]\mapsto[ax_1+bx_0,cx_1+dx_0]$. And that's essentially a linear map between vectors, since these coordinates originally refer back to a point in $\mathbb R^2$ (points on the lines through which we project). And the matrix corresponding to that map is

$$\begin{pmatrix}a&b\\c&d\end{pmatrix}.$$

So fractional linear transformations essential act as linear transformations on the homogenous coordinates, which can then be translated to points on the projective line. Now if we compose fractional linear maps, we're essentially composing linear maps on the coordinate space $\mathbb R^2$. And that's done via matrix multiplication. So we should expect to find that composition and inversion of fractional linear maps somehow mimics matrix multiplication and inversion.

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  • $\begingroup$ The explanation seems very technical, is it possible to dilute it a bit with simpler terminology and/or pictures? $\endgroup$ Commented Dec 11, 2021 at 11:09
  • $\begingroup$ I know. I wasn't too happy with it either, but wrote it on mobile, where it's a pain to find and add good pictures. $\endgroup$ Commented Dec 11, 2021 at 11:24
  • $\begingroup$ Yes, I would appreciate both simpler terminology and pictures. $\endgroup$
    – Starlight
    Commented Dec 11, 2021 at 13:03
  • $\begingroup$ I added a picture, @Starlight $\endgroup$ Commented Dec 11, 2021 at 13:50
  • $\begingroup$ Thanks. That is useful. $\endgroup$
    – Starlight
    Commented Dec 11, 2021 at 15:04

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