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arXiv · 2607.05589

Powers and trace of symmetric powers of $2\times 2$ matrices and combinatorial, Fibonacci and Lucas identities

José Luis Cisneros-Molina
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Abstract

Let $A$ be an arbitrary $2\times 2$ matrix. In \cite{Cisneros:PhD,Cisneros:I2x2M} I gave a formula for the trace of the $k$-th symmetric power of $A$ in terms of the anti-diagonal entries of $A^{k+1}$ and $A$. This was based on formulae that I found for the entries of the $k$-th power $A^k$ of the matrix $A$ in terms of its entries but I only sketched the idea of how I obtained such formulae. In this article I give the full proof of those formulae by counting some walks of length $k$ over the complete digraph of order $2$. I compare them with formulae for $A^k$ given by Mc Laughlin in \cite{McLaughlin:CIDnP2x2M} and by Williams in \cite{Williams:nthP2x2M}. This leads to combinatorial identities, in particular expressions for Fibonacci and Lucas numbers.

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José Luis Cisneros-Molina. 2026-07-06. Powers and trace of symmetric powers of $2\times 2$ matrices and combinatorial, Fibonacci and Lucas identities. https://arxiv.org/abs/2607.05589

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