On the principal minors of the powers of a matrix

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.

13 pages. Note inspired by grading the Putnam in 2021. v2 fixes various minor errors and adds a forward-reference to the answer of the question from Section 6