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 .

[To appear in the Palestinian Journal of Mathematics]

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

## Sample Input and Output files for Larcombe.txt

• If you want to see the all relations (up to trivial isomorphism) between n non-diagonal entries in an arbitrary power of a generic n by n matrix for n=2 (the original Larcombe-Fennessey case) and n=3, n=4

the input file yields the output file

• Added Aug. 8, 2021: Inspired by a question of Peter Larcombe, we added a new procedure to the Maple package called NuLarcombeRel(N). It outputs the first N terms of the sequence "Number of INEQUIVALENT Larcombe relations for n by n matrices", equivalently, the number of UNLABELLED loopless DIRECTED graphs on n vertices and having n edges.

the input file yields the output file

Note: This sequence was not (Aug. 8, 2021) in the OEIS. On the other hand the total number of such graphs is in the OEIS.

[It uses Polya-Redfield theory]

• [Added Sept. 4, 2021: Peter Larcombe, asked whether there is any hope for finding Larcombe relations for dimensions higher than four. Well, for numeric matrices can you go very far. We chose ramdom matrices for dimensions 5 through 10.

The input file yields the output file

Doron Zeilberger's papers