Möbius transformations are the fractional linear changes of variable that preserve the projective geometry of the complex line. They sit at the meeting point of complex analysis, linear algebra, [group actions](/page/Group%20Action), and the geometry of the Riemann sphere. Their defining formula is short, but it encodes several large facts: the [automorphisms of the Riemann sphere](/theorems/7862), the symmetries of many planar domains, the invariance of the cross-ratio, and the way lines and circles form one unified family.

The naive class of affine maps $z \mapsto az+b$ is too small for the natural compactification of the complex plane. Once the point at infinity is included, inversion becomes as basic as translation and rotation:

\begin{align*} z &\mapsto \frac{1}{z}. \end{align*}

Möbius transformations are the family generated under composition by translations, dilations, rotations, and inversion; equivalently, they are the projective linear group $PGL(2,\mathbb{C})$ acting on the complex projective line $\mathbb{CP}^1$. Here $PGL(2,\mathbb{C})$ means the group of invertible $2\times 2$ complex matrices after identifying matrices that differ by a nonzero scalar multiple, and $\mathbb{CP}^1$ is the set of one-dimensional complex lines in $\mathbb{C}^2$, naturally identified with the Riemann sphere.

## Definition

The main object is a fractional [linear map](/page/Linear%20Map) that is allowed to send an ordinary complex number to infinity and to have a value at infinity itself. Temporarily write $\widehat{\mathbb{C}}$ for the complex plane enlarged by one point. The determinant condition below is the algebraic test that the fractional expression is not a disguised constant map.

[definition: Mobius Transformation] A Möbius transformation is a map $T: \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$ for which there exist $a,b,c,d \in \mathbb{C}$ satisfying \begin{align*} ad-bc \ne 0 \end{align*} and \begin{align*} T(z) = \frac{az+b}{cz+d} \end{align*} for every $z \in \mathbb{C}$ with $cz+d \ne 0$, extended to the exceptional finite point and to $\infty$ as a map on $\widehat{\mathbb{C}}$. [/definition]

The extension rule is part of the construction: if $c \ne 0$, then $T(-d/c)=\infty$ and $T(\infty)=a/c$; if $c=0$, then $T(\infty)=\infty$.

To make those assignments mathematical rather than informal, the symbol $\widehat{\mathbb{C}}$ has to denote a genuine space. The point at infinity is not a decorative extra point: it is what turns the pole of a fractional expression into an ordinary value and lets these maps be studied globally on the same compact surface as the Riemann sphere.

[definition: Extended Complex Plane] The extended complex plane is the set \begin{align*} \widehat{\mathbb{C}} := \mathbb{C} \cup \{\infty\}. \end{align*} It is equipped with the topology of the Riemann sphere. [/definition]

A practical difficulty appears as soon as the coefficients are written down: multiplying all of them by the same nonzero scalar changes no value of the map. The next observation records this ambiguity before we use matrices to organise composition.

[remark: Scalar Ambiguity] If $\lambda \in \mathbb{C}\setminus\{0\}$, then the coefficient lists $(a,b,c,d)$ and $(\lambda a,\lambda b,\lambda c,\lambda d)$ define the same Möbius transformation. [/remark]

The formula needs a bookkeeping device that remembers composition but forgets irrelevant scalar multiples. The projective matrix representation supplies exactly that device: it turns Möbius transformations into matrix classes rather than isolated fractional expressions.

[definition: Projective Matrix of a Mobius Transformation] A projective matrix representing a Möbius transformation is an equivalence class $[A]$ of invertible matrices $A \in GL(2,\mathbb{C})$ under the relation $A \sim \lambda A$ for $\lambda \in \mathbb{C}\setminus\{0\}$. If $A$ has first row $(a,b)$ and second row $(c,d)$, then $[A]$ determines the map $T_A: \widehat{\mathbb{C}} \to \widehat{\mathbb{C}}$ defined by \begin{align*} T_A(z)=\frac{az+b}{cz+d} \end{align*} for $z \in \mathbb{C}$ with $cz+d \ne 0$, with the corresponding extended values at the pole and at $\infty$. [/definition]

The projective viewpoint is the bridge from formulas to geometry. It packages the fractional expression in a way that remembers composition while still identifying scalar multiples that give the same map. The same viewpoint will reappear later when composition, inverses, and domain automorphisms are treated as consequences of one projective action rather than as unrelated coordinate tricks.

[example: Translation as a Projective Matrix] The translation $T(z)=z+2$ is represented by the matrix with first row $(1,2)$ and second row $(0,1)$: \begin{align*} \begin{pmatrix}1&2\end{pmatrix} \quad\text{above}\quad \begin{pmatrix}0&1\end{pmatrix}. \end{align*} Indeed, for finite $z$, \begin{align*} \frac{1\cdot z+2}{0\cdot z+1}=z+2, \end{align*} and the determinant is $1$. This small example shows why affine maps are already Möbius transformations: the fractional formula includes them as the case $c=0$. [/example]

## Equivalent Characterisations

The most concrete definition uses a fractional formula, but the same class can be recognised intrinsically. This matters because in complex analysis one often meets a map first as a holomorphic bijection, not as a displayed quotient of linear functions.

The intrinsic question is now whether the fractional formula is merely one convenient way to produce symmetries of the sphere, or whether it captures all of them. A holomorphic automorphism of the Riemann sphere is first encountered as a global analytic bijection, with no coefficients displayed. The next theorem closes that gap by showing that the analytic and projective descriptions define the same class of maps.

[quotetheorem:7862]