Powers of matrices with all principal minors equal to 1

Abstract

We say that a square matrix $A$ is \emph{$1$-principled} if all its principal minors are equal to $1$. We show that over any well-behaved ring, any power $A^m$ of a $1$-principled matrix $A$ is again $1$-principled. Well-behaved rings include all reduced rings as well as all quotients of commutative rings modulo integrally closed ideals; in particular, all fields and all quotients of $\mathbb{Z}$ are well-behaved. We note that $m$ can be any integer, positive or negative. This generalizes Problem B5 of the 2021 Putnam contest in multiple directions. Over arbitrary commutative rings, we identify a stronger property that is always inherited by powers: We say that a matrix $A = \left(a_{i,j}\right)_{i,j\in\left[n\right]}$ is \emph{$1$-nullcyclic} if all its diagonal entries are $1$ and if all the cyclic products $a_{i_1, i_2} a_{i_2, i_3} \cdots a_{i_k, i_1}$ with $k>1$ and distinct $i_1,i_2,\ldots,i_k$ vanish. We show that if $A$ is $1$-nullcyclic, then so is $A^m$ for any integer $m$. Furthermore, every $1$-nullcyclic matrix is $1$-principled over any commutative ring, while the converse holds if the ring is well-behaved. Along the way, we prove analogous results that don't require the diagonal entries to be $1$. These are concerned with \emph{principled matrices} (those whose principal minors equal the respective products of diagonal entries) and \emph{nullcyclic matrices} (those whose cyclic products $a_{i_1, i_2} a_{i_2, i_3} \cdots a_{i_k, i_1}$ with $k>1$ and distinct $i_1,i_2,\ldots,i_k$ vanish); their diagonal entries can be arbitrary. A crucial auxiliary result, which holds for any $n\times n$-matrix $A$, is that the cyclic products $a_{i_1, i_2} a_{i_2, i_3} \cdots a_{i_k, i_1}$ (with $k>1$ and distinct $i_1,i_2,\ldots,i_k$) are integral over the ideal generated by the principal minors of $A$ minus the corresponding products of diagonal entries of $A$.