Two Theorems in Multi-Weighted Sums

Abstract

Systems of linear relationships have received an increasing amount of attention in recent years, partly in their own right as mathematical entities, but probably more so because of their frequent occurrence in physical situations. Multilinear systems, however, involving linear combinations of linear expressions, seem to have been almost completely neglected. Yet many physical situations are more adequately described by multilinear than by linear equations. Examples arise in the topological synthesis of electrical networks, and in sensitivity and stability studies of active networks. There is therefore a need for mathematical development in this area. To the end of initiating this, two theorems, based on the representation of a multilinear system as a multi-weighted sum of vectors belonging to some linear space, are presented in this paper, together with an example of their application in the stability analysis of linear systems.