Skew-symmetric matrices and their principal minors

Abstract

Let $V$ be a nonempty finite set and $A=(a_{ij})_{i,j\in V}$ be a matrix with entries in a field $\mathbb{K}$. For a subset $X$ of $V$, we denote by $A[X]$ the submatrix of $A$ having row and column indices in $X$. We study the following problem. Given a positive integer $k$, what is the relationship between two matrices $A=(a_{ij})_{i,j\in V}$, $B=(b_{ij})_{i,j\in V}$ with entries in $\mathbb{K}$ and such that $\det(A\left[ X\right])=\det(B\left[ X\right])$ for any subset $X$ of $V$ of size at most $k$ ? The Theorem that we get in this Note is an improvement of a result of R. Loewy [5] for skew-symmetric matrices whose all off-diagonal entries are nonzero.