On the principal minors of the powers of a matrix

Darij Grinberg

On the principal minors of the powers of a matrix

Not yet reviewed by Pith; the record is open.

Re-run · record.json Download PDF Read on arXiv ↗

This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.

SPECIMEN: schema-true, not a live event

T0 review · schema-true

One-sentence machine reading of the paper's core claim.

pith:XXXXXXXX · record.json · timestamp

arxiv 2204.07885 v2 pith:VM2WTEIT submitted 2022-04-16 math.RA

Darij Grinberg This is my paper · ORCID

classification math.RA

keywords matrixminorslatterprincipalthenbehaviorcontestdetermined

verification ladder T0 review T1 audit T2 compute T3 formal

0 comments

abstract

We show that if $A$ is an $n\times n$-matrix, then the diagonal entries of each power $A^{m}$ are uniquely determined by the principal minors of $A$, and can be written as universal (integral) polynomials in the latter. Furthermore, if the latter all equal $1$, then so do the former. These results are inspired by Problem B5 on the Putnam contest 2021, and shed a new light on the behavior of minors under matrix multiplication.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Powers of matrices with all principal minors equal to 1
math.AC 2026-06 unverdicted novelty 7.0 of 10

Powers preserve the all-principal-minors-equal-to-1 property over reduced rings, Z/d and integrally closed quotients, via a nullcyclic strengthening that works over every commutative ring.

Pith tools