On the principal minors of the powers of a matrix

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.