Matrices in the Hosoya triangle

TL;DR

This paper investigates properties of matrices derived from the Hosoya triangle using linear algebra, revealing Fibonacci identities, eigenbehavior, and graph structures mod 2, blending algebraic and geometric insights.

Contribution

It introduces a novel analysis of matrices in the Hosoya triangle, connecting linear algebra, Fibonacci identities, and graph theory in a unified framework.

Findings

Eigenvalues and eigenvectors relate to Fibonacci identities.

Most matrix properties embed again in the Hosoya triangle.

Identified an infinite family of non-connected graphs with specific structures.

Abstract

In this paper we use well-known results from linear algebra as tools to explore some properties of products of Fibonacci numbers. Specifically, we explore the behavior of the eigenvalues, eigenvectors, characteristic polynomials, determinants, and the norm of non-symmetric matrices embedded in the Hosoya triangle. We discovered that most of these objects either embed again in the Hosoya triangle or they give rise to Fibonacci identities. We also study the nature of these matrices when their entries are taken $\bmod$ $2$. As a result, we found an infinite family of non-connected graphs. Each graph in this family has a complete graph with loops attached to each of its vertices as a component and the other components are isolated vertices. The Hosoya triangle allowed us to show the beauty of both, the algebra and geometry.