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Decompositions of linear operators on pre-euclidean spaces by means of graphs

Hani Abdelwahab, Elisabete Barreiro, Antonio J. Calderón, José M. Sánchez

Abstract

In this work we study a linear operator $f$ on a pre-euclidean space $\mathcal{V}$ by using properties of a corresponding graph. Given a basis $\B$ of $\mathcal{V}$, we present a decomposition of $\mathcal{V}$ as an orthogonal direct sum of certain linear subspaces $\{U_i\}_{i \in I}$, each one admitting a basis inherited from $\B$, in such way that $f = \sum_{i \in I}f_i$, being each $f_i$ a linear operator satisfying certain conditions respect with $U_i$. Considering new hypothesis, we assure the existence of an isomorphism between the graphs associated to $f$ relative to two different bases. We also study the minimality of $\mathcal{V}$ by using the graph associated to $f$ relative to $\B$.

Decompositions of linear operators on pre-euclidean spaces by means of graphs

Abstract

In this work we study a linear operator on a pre-euclidean space by using properties of a corresponding graph. Given a basis of , we present a decomposition of as an orthogonal direct sum of certain linear subspaces , each one admitting a basis inherited from , in such way that , being each a linear operator satisfying certain conditions respect with . Considering new hypothesis, we assure the existence of an isomorphism between the graphs associated to relative to two different bases. We also study the minimality of by using the graph associated to relative to .
Paper Structure (6 sections, 7 theorems, 38 equations)

This paper contains 6 sections, 7 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Linear operator on a pre-euclidean space and graphs. Decomposition Theorem
  4. Relating the graphs given by different choices of bases
  5. Characterization of the minimality and weak symmetry
  6. Conclusion

Key Result

Theorem 3.1

Let $f : \mathcal{V} \to \mathcal{V}$ be a linear operator on a pre-euclidean space $(\mathcal{V}, \langle \cdot , \cdot \rangle)$ with basis $\mathcal{B}=\{e_i\}_{i \in I}$. Then the following statements are equivalent.

Theorems & Definitions (33)

  • Definition 2.1
  • Example 2.1
  • Definition 2.2
  • Definition 3.1
  • Example 3.1
  • Example 3.2
  • Definition 3.2
  • Definition 3.3
  • Example 3.3
  • Example 3.4
  • ...and 23 more