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Let $\mathbb{K}$ denote a field and let $\mathfrak{X}$ denote a finitenon-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 existmutually 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 algebraiccharacterization of a Cauchy matrix. To do so, we introduce the notion of aCauchy pair. A Cauchy pair is an ordered pair of diagonalizable lineartransformations $(X, \tilde{X})$ on a finite-dimensional vector space $V$ suchthat $X-\tilde{X}$ has rank 1 and such that there does not exist a propersubspace $W$ 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 thetransition 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 atransition matrix for a Cauchy pair in this way. We give a bijection betweenthe set of equivalence classes of Cauchy pairs on $V$ and the set ofpermutation equivalence classes of Cauchy matrices in$\text{Mat}_\mathfrak{X}(\mathbb{K})$.

Cauchy Pairs and Cauchy Matrices