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arXiv · 2204.07885

On the principal minors of the powers of a matrix

Darij Grinberg
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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.

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Darij Grinberg. 2022-04-16. On the principal minors of the powers of a matrix. https://arxiv.org/abs/2204.07885

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