The Spectral Method of Calculation of Symmetric Structures of Finite Size

Abstract

This article suggests a method of computing linear problems for multi-element symmetric structures of finite size; i.e., the multi-element structures associated with symmetry groups of order h > 1. Every such symmetric structure corresponds to a symmetric matrix which may have other matrices as its elements; we call the original matrix the “block-matrix” and its elements “submatrices”. In theory, every symmetric matrix can be diagonalized. In practice, however, the process of numerical diagonalization may require a prohibitive amount of computer time. If a combination of analytical and numerical decompositions of such matrices is available, it can be then utilized to substantially reduce the amount of calculation and the requirement of computer time. One such method, which we call the “Spectral Method”, is offered herein. This method is subsequently used to solve both homogeneous and nonhomogeneous systems of linear equations and to invert corresponding matrices of such structures.