Tridiagonal real symmetric matrices with a connection to Pascal's triangle and the Fibonacci sequence

TL;DR

This paper investigates a family of symmetric tridiagonal matrices, deriving their characteristic polynomials, revealing connections to Pascal's triangle and Fibonacci numbers, and exploring eigenvalue inclusion relations.

Contribution

It provides a closed-form solution for the characteristic polynomials of these matrices and establishes conditions for eigenvalue inclusion, linking combinatorics and spectral theory.

Findings

Characteristic polynomials involve Pascal's triangle coefficients

Eigenvalue inclusion conditions are established

Connections to Fibonacci sequence are proposed

Abstract

We explore a certain family $\{A_n\}_{n=1}^{\infty}$ of $n \times n$ tridiagonal real symmetric matrices. After deriving a three-term recurrence relation for the characteristic polynomials of this family, we find a closed form solution. The coefficients of these characteristic polynomials turn out to involve the diagonal entries of Pascal's triangle in a tantalizingly predictive manner. Lastly, we explore a relation between the eigenvalues of various members of the family. More specifically, we give a sufficient condition on the values $m,n \in \mathbb{N}$ for when $\texttt{spec}(A_m)$ is contained in $\texttt{spec}(A_n)$. We end the paper with a number of open questions, one of which intertwines our characteristic polynomials with the Fibonacci sequence in an intriguing manner involving ellipses.