Cartan–Dieudonné theorem

In mathematics, the Cartan–Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, is an important idea that helps us understand certain kinds of geometric transformations. This theorem tells us that every special kind of change, called an orthogonal transformation, in a space with n dimensions can be created by linking together, or composing, at most n simple actions called reflections.

The spaces we are talking about here are called symmetric bilinear spaces, which are like generalizations of the flat, straight-line spaces we are used to. These spaces are defined by something called a symmetric bilinear form, which does not always have to be positive, meaning these spaces can be more complex than regular Euclidean space. For example, a pseudo-Euclidean space is also a symmetric bilinear space.

In everyday 2D space, like the flat surface of a table, any change that keeps distances and angles unchanged can either be a flip over a line through a point or a turn around that point. Remarkably, any combination of these flips and turns can always be simplified to just two flips linked together. In 3D space, such as our world, any of these changes can be a single flip, a turn (which is two flips linked), or a special kind of turn called an improper rotation (which uses three flips). In four dimensions, there are even more complex changes called double rotations that need four flips to describe.

Formal statement

In a special kind of mathematical space with certain rules, every transformation that keeps distances and angles the same can be made by combining a small number of simple flips. Specifically, you only need up to n of these flips to get any such transformation in an n-dimensional space.