Groups of Congruences and Restriction Matrices

Author： Aimi A. Bassotti L. Diligenti M.

ISSN： 0006-3835

Source： Bit Numerical Mathematics, Vol.43, Iss.4, 2003-12, pp. : 671-693

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Abstract

Let Ω be a domain of the Euclidean space R^m sent onto itself by a finite group G of congruences. In this paper we first define m elementary restriction matrices related to the group G and to a system of irreducible matrix representations of G. We then describe a general procedure to generate m restriction matrices for any finite-dimensional space V(Ω) of real functions defined on Ω, when V(Ω) is invariant with respect to G. The number m depends only on the group G. Restriction matrices for the space V(Ω) have a block structure and all blocks can be obtained as from an elementary restriction matrix. Restriction matrices related to V(Ω) define a decomposition of V(Ω) as the sum of m subspaces. Finally, owing to restriction matrices, we propose a result of decomposition for linear systems. Several examples are presented.