Recognizing matrices

Rotation matrices and rotoreflection matrices, in numerical form, usually have no distinguishing characteristic identifiable by inspection. Unless they are especially simple, they look just about like any other 3-by-3 real numerical matrix filled with values between −1 and +1. In this chapter we develop numerical tests to distinguish them, and to extract the axis and the angle. These tests are collected together in a useful operator named RecognizeMatrix, based on the following theorem