On Invariance Properties of Entries of Matrix Powers
Combinatorics
2021-07-29 v1
Abstract
A few years ago, Peter Larcombe discovered an amazing property regarding two by two matrices. For any such 2 by 2 matrix A, the ratios of the two anti-diagonal entries is the same for all powers of A. We discuss extensions to higher dimensions, and give a short bijective proof of Larcombe and Eric Fennessey's elegant extension to tri-diagonal matrices of arbitrary dimension. This article is accompanied by a Maple package.
Keywords
Cite
@article{arxiv.2107.13092,
title = {On Invariance Properties of Entries of Matrix Powers},
author = {Shalosh B. Ekhad and Doron Zeilberger},
journal= {arXiv preprint arXiv:2107.13092},
year = {2021}
}
Comments
6 pages, accompanied by a Maple package and output file obtainable from https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/larcombe.html