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arXiv · 2201.08490

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

Emily Gullerud·Rita Johnson·aBa Mbirika
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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.

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Emily Gullerud, Rita Johnson, aBa Mbirika. 2022-01-21. Tridiagonal real symmetric matrices with a connection to Pascal's triangle and the Fibonacci sequence. https://arxiv.org/abs/2201.08490

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