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

Hani Abdelwahab, Elisabete Barreiro, Antonio J. Calderón and José M. Sánchez ()
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Hani Abdelwahab: Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Elisabete Barreiro: University of Coimbra, CMUC, Department of Mathematics, FCTUC, Largo D. Dinis, 3000-143 Coimbra, Portugal
Antonio J. Calderón: Department of Mathematics, University of Cádiz, 11510 Puerto Real, Spain
José M. Sánchez: Department of Mathematics, University of Cádiz, 11510 Puerto Real, Spain

Mathematics, 2023, vol. 11, issue 3, 1-12

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 { U i } i ∈ I , each one admitting a basis inherited from B , in such way that f = ∑ i ∈ I f i . Each f i is a linear operator satisfying certain conditions with respect to U i . 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 .

Keywords: linear operators; pre-Euclidean spaces; graph theory (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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