On Invariance Properties of Entries of Matrix Powers

By Shalosh B. Ekhad and Doron Zeilberger

.pdf   .tex  

First Written: July 27, 2021 ; This version: Aug. 4, 2021 .

[appeared in the Palestinian Journal of Mathematics Vol. 11(2), (2022), 1-5)]

A few years ago, Peter Larcombe discovered an amazing property regarding two by two matrices. For any such 2 times 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.

Added Sept. 29, 2021: Read the insightful comments by Darij Grinberg.

Added Aug. 4, 2021: With Darij Grinberg's kind permission, this is now included in the actual article.

Added Sept. 22, 2021: Sajal Kumar Mukherjee and Sanjay Mukherjee just wrote a beautiful and far-reaching generalization of Larcombe-Fennesey relations. Strongly remommended!

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