On the principal minors of the powers of a matrix

TL;DR

This paper proves that the diagonal entries of matrix powers are uniquely determined by the principal minors of the original matrix and can be expressed as universal polynomials, with implications for matrix behavior analysis.

Contribution

It establishes a novel relationship between principal minors and diagonal entries of matrix powers, inspired by a Putnam problem, providing new insights into minors' behavior under multiplication.

Findings

Diagonal entries of $A^m$ are determined by principal minors of $A$

Diagonal entries can be expressed as universal polynomials in principal minors

If all principal minors are 1, diagonal entries of powers are also 1

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.