Canonical Forms, Basic Definitions and Properties

Abstract

In this chapter, linear algebra is employed for the study of symmetry, followed by graph-theoretical interpretations. The application of the methods presented in this chapter is not limited to geometric symmetry. Thus, the symmetry studied here can more appropriately be considered as topological symmetry. The methods considered in this chapter can be considered as special techniques for transforming the matrices into block triangular forms. These forms allow good saving of computation effort for many important problems such as computing determinants, eigenvalue problems and solution of linear system of equations. For each of these tasks with dimension N, the computing cost grows approximately with N³. Therefore, reducing, for example, the dimension to N/2, the effort decreases eight times which is a great advantage.