Linear Algebra, 5th Edition

6.11* The Geometry of Orthogonal Operators

By Theorem 6.22 (p. 383), any rigid motion on a finite-dimensional real inner product space is the composite of an orthogonal operator and a translation. Thus, to understand the geometry of rigid motions thoroughly, we must analyze the structure of orthogonal operators. In this section, we show that any orthogonal operator on a finite-dimensional real inner product space can be described in terms of rotations and reflections.

This material assumes familiarity with the results about direct sums developed at the end of Section 5.2 and with the definition of the determinant of a linear operator given in Section 5.1 as well as elementary properties of the determinant in Exercise 8 of Section 5.1.

We now ...