Cauchy Pairs and Cauchy Matrices

Abstract

Let $\mathbb{K}$ denote a field and let $\mathfrak{X}$ denote a finite non-empty set. Let $\text{Mat}_\mathfrak{X}(\mathbb{K})$ denote the $\mathbb{K}$-algebra consisting of the matrices with entries in $\mathbb{K}$ and rows and columns indexed by $\mathfrak{X}$. A matrix $C \in \text{Mat}_\mathfrak{X}(\mathbb{K})$ is called Cauchy whenever there exist mutually distinct scalars $\{x_i\}_{i \in \mathfrak{X}}, \{\tilde{x}_i\}_{i \in \mathfrak{X}}$ from $\mathbb{K}$ such that $C_{ij} = (x_i - \tilde{x}_j)^{-1}$ for $i, j \in \mathfrak{X}$. In this paper, we give a linear algebraic characterization of a Cauchy matrix. To do so, we introduce the notion of a Cauchy pair. A Cauchy pair is an ordered pair of diagonalizable linear transformations $(X, \tilde{X})$ on a finite-dimensional vector space $V$ such that $X-\tilde{X}$ has rank 1 and such that there does not exist a proper subspace $W$ of $V$ such that $X W \subseteq W$ and $\tilde{X} W \subseteq W$. Let $V$ denote a vector space over $\mathbb{K}$ with dimension $|\mathfrak{X}|$. We show that for every Cauchy pair $(X, \tilde{X})$ on $V$, there exists an $X$-eigenbasis $\{v_i\}_{i \in \mathfrak{X}}$ for $V$ and an $\tilde{X}$-eigenbasis $\{w_i\}_{i \in \mathfrak{X}}$ for $V$ such that the transition matrix from $\{v_i\}_{i \in \mathfrak{X}}$ to $\{w_i\}_{i \in \mathfrak{X}}$ is Cauchy. We show that every Cauchy matrix arises as a transition matrix for a Cauchy pair in this way. We give a bijection between the set of equivalence classes of Cauchy pairs on $V$ and the set of permutation equivalence classes of Cauchy matrices in $\text{Mat}_\mathfrak{X}(\mathbb{K})$.