Finite symmetric systems and their analysis (Q1203957)

This paper is addressed to the classification and structural analysis of the finite linear symmetric systems. The systems are defined in terms of symmetry operations as rotations about axes, reflections in planes and translations. They are classified with respect to their symmetry elements. The so-called algebraic approach used herein is based on the explicit block diagonal decomposition of the matrix equation corresponding to a symmetric system. Symmetry matrices corresponding to the cyclic symmetric system are introduced and their fundamental properties are discussed. Also, matrices corresponding to the finite linear symmetric systems are constructed and related theorems are proved. The solutions of linear equations introduced on the finite symmetric systems are given and the efficiency of the method is compared with the Gaussian elimination. It is shown that the efficiency depends on symmetry of the applied load, and it substantially increases if the loads are symmetric. A sample example is provided for the structural analysis of a vacuum vessel of a fusion machine subjected to a set of loads with different types of symmetry. Group theory and especially group representation theory are widely used throughout the paper. This interesting study is theoretical in nature and it seems to have fruitful applications.