the (mutually reciprocal) eigenvalue ratios of the matrix representing the transform (compare the discussion in the preceding section about the characteristic constant of a transformation). Its characteristic polynomial is equal to which has roots Simple Möbius transformations and composition A Möbius transformation can be composed as a sequence of simple transformations. The following simple transformations are also Möbius transformations: is a translation. is a combination of a (homothety and a rotation). If then it is a rotation, if then it is a homothety. (inversion and reflection with respect to the real axis) Composition of simple transformations If , let: (translation by d/c) (inversion and reflection with respect to the real axis) (homothety and rotation) (translation by a/c) Then these functions can be composed, showing that, if one has In other terms, one has with This decomposition makes many properties of the Möbius transformation obvious. Elementary properties A Möbius transformation is equivalent to a sequence of simpler transformations. The composition makes many properties of the Möbius transformation obvious. Formula for the inverse transformation The existence of the inverse Möbius transformation and its explicit formula are easily derived by the composition of the inverse functions of the simpler transformations. That is, define functions g1, g2, g3, g4 such that each gi is the inverse of fi. Then the composition gives a formula for the inverse. Preservation of angles and generalized circles From this decomposition, we see that Möbius transformations carry over all non-trivial properties of circle inversion. For example, the preservation of angles is reduced to proving that circle inversion preserves angles since the other types of transformations are dilations and isometries (translation, reflection, rotation), which trivially preserve angles. Furthermore, Möbius transformations map generalized circles to generalized circles since circle inversion has this property. A generalized circle is either a circle or a line, the latter being considered as a circle through the point at infinity. Note that a Möbius transformation does not necessarily map circles to circles and lines to lines: it can mix the two. Even if it maps a circle to another circle, it does not necessarily map the first circle's center to the second circle's center. Cross-ratio preservation Cross-ratios are invariant under Möbius transformations. That is, if a Möbius transformation maps four distinct points to four distinct points respectively, then If one of the points is the point at infinity, then the cross-ratio has to be defined by