The Cartan-Dieudonné Theorem

Abstract

In this chapter the structure of the orthogonal group is studied in more depth. In particular, we prove that every isometry in O(n) is the composition of at most n reflections about hyperplanes (for n ≥ 2, see Theorem 7.2.1). This important result is a special case of the "Cartan-Dieudonné theorem" (Cartan [29], Dieudonné [47]). We also prove that every rotation in SO(n) is the composition of at most n flips (for n ≥ 3). Hyperplane reflections are represented by matrices called Householder matrices. These matrices play an important role in numerical methods, for instance for solving systems of linear equations, solving least squares problems, for computing eigenvalues, and for transforming a symmetric matrix into a tridiagonal matrix. We prove a simple geometric lemma that immediately yields the QR-decomposition of arbitrary matrices in terms of Householder matrices. Affine isometries are defined, and their fixed points are investigated. First, we characterize the set of fixed points of an affine map. Using this characterization, we prove that every affine isometry f can be written uniquely as f = t • g, with t • g = g • t, where g is an isometry having a fixed point, and t is a translation by a vector τ such that − → f (τ) = τ , and with some additional nice properties (see Lemma 7.6.2). This is a generalization of a classical result of Chasles