On the decomposition of vector spaces

Abstract

This paper attempts to put the notion of “network decomposition into multiports” in electrical network theory on a rigorous mathematical footing. This is done by first defining the notion of decomposition for vector spaces. The given vector space V, defined on S, is expressed as (⊕ V A i P i + V P ) × S, where V A i P i are defined on A i ∪ P i with ⋃ · A i = S , and V P is defined on ⋃ · P i . The following questions are examined: When can given V A i P i be coupled to make up a specified V S ? When is the decomposition minimal (i.e., when is | P| minimum, A i being specified), and what are its characteristic properties? How to build a minimal decomposition? Can a minimal decomposition be built preserving certain matroid properties going from the original to the decomposition (such as decomposing a graph into graphs)? Can one recover the matroid of the original vector space, knowing the matroids of the decomposition? It is shown that operations on the original vector space such as contraction, restriction, and dualization can be performed indirectly by using the same operation on the decomposition. Finally, applications (which in fact gave rise to the theory) to network analysis are described.