Linear fractional transformation

In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form which has an inverse. The precise definition depends on the nature of a, b, c, d, and z. In other words, a linear fractional transformation is a transformation that is represented by a fraction whose numerator and denominator are linear. In the most basic setting, a, b, c, d, and z are complex numbers (in which case the transformation is also called a Möbius transformation), or more generally elements of a field. The invertibility condition is then ad – bc ≠ 0. Over a field, a linear fractional transformation is the restriction to the field of a projective transformation or homography of the projective line. When a, b, c, d are integer (or, more generally, belong to an integral domain), z is supposed to be a rational number (or to belong to the field of fractions of the integral domain. In this case, the invertibility condition is that ad – bc must be a unit of the domain (that is 1 or −1 in the case of integers). In the most general setting, the a, b, c, d and z are square matrices, or, more generally, elements of a ring. An example of such linear fractional transformation is the Cayley transform, which was originally defined on the 3 x 3 real matrix ring. Linear fractional transformations are widely used in various areas of mathematics and its applications to engineering , such as classical geometry, number theory (they are used, for example, in Wiles's proof of Fermat's Last Theorem), group theory, control theory. In general, a linear fractional transformation is a homography of P(A), the projective line over a ring A. When A is a commutative ring, then a linear fractional transformation has the familiar form where a, b, c, d are elements of A such that ad – bc is a unit of a (that is ad – bc has a multiplicative inverse in A)