Cartan–Dieudonné theorem

In mathematics, the Cartan–Dieudonné theorem is a key idea named after Élie Cartan and Jean Dieudonné. It helps us understand special changes in geometric spaces.

This theorem says that any change called an orthogonal transformation in a space with n dimensions can be made by linking together, or composing, at most n simple actions called reflections.

The spaces discussed are called symmetric bilinear spaces. These are like generalizations of flat, straight-line spaces we know. They are defined by something called a symmetric bilinear form, which can be more complex than regular flat 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 be a flip over a line through a point or a turn around that point. 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 math space, every change that keeps distances and angles the same can be made by combining a small number of simple flips. You only need up to n flips to make any such change in an n-dimensional space.