Cauchy pairs and Cauchy matrices

Let K denote a field and let X denote a finite non-empty set. Let MatX(K) denote the K-algebra consisting of the matrices with entries in K and rows and columns indexed by X. A matrix C∈MatX(K) is called Cauchy whenever there exist mutually distinct scalars {xi}i∈X,{x˜i}i∈X from K such that Cij=(xi−x˜j)−1 for i,j∈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,X˜) on a finite-dimensional vector space V such that X−X˜ has rank 1 and such that there does not exist a proper subspace W of V such that XW⊆W and X˜W⊆W. Let V denote a vector space over K with dimension |X|. We show that for every Cauchy pair (X,X˜) on V, there exists an X-eigenbasis {vi}i∈X for V and an X˜-eigenbasis {wi}i∈X for V such that the transition matrix from {vi}i∈X to {wi}i∈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 MatX(K).