Decomposition of Linear Operators on Pre-Euclidean Spaces by Means of Graphs

Abstract

In this work, we study a linear operator f on a pre-Euclidean space V by using properties of a corresponding graph. Given a basis B of V, we present a decomposition of V as an orthogonal direct sum of certain linear subspaces {Ui}i∈I, each one admitting a basis inherited from B, in such way that f=∑i∈Ifi. Each fi is a linear operator satisfying certain conditions with respect to Ui. Considering this new hypothesis, we assure the existence of an isomorphism between the graphs of f relative to two different bases. We also study the minimality of V by using the graph of f relative to B.