柯西对与柯西矩阵
Cauchy Pairs and Cauchy Matrices
摘要：
设$\mathbb{K}$表示域，$\mathfrak{X}$表示有限非空集.令$\text{Mat}_\mathfrak{X}(\mathbb{K})$表示$\mathbb{K}$ -代数，其由具有$\mathbb{K}$中的元素的矩阵以及由$\mathfrak{X}$索引的行和列组成。矩阵$C \in \text{Mat}_\mathfrak{X}(\mathbb{K})$称为柯西，只要存在与$\mathbb{K}$相互不同的标量$\{x_i\}_{i \in \mathfrak{X}}, \{\tilde{x}_i\}_{i \in \mathfrak{X}}$，使得$i, j \in \mathfrak{X}$的$C_{ij} = (x_i - \tilde{x}_j)^{-1}$。本文给出了柯西矩阵的线性代数特征。为此，我们引入柯西对的概念。柯西对是有限维向量空间$V$上的可对角化线性变换$(X, \tilde{X})$的有序对，使得$X-\tilde{X}$具有秩1，并且不存在$V$的真子空间$W$，使得$X W \subseteq W$和$\tilde{X} W \subseteq W$。令$V$表示$\mathbb{K}$上的具有维度$|\mathfrak{X}|$的向量空间。我们证明了对于$V$上的每个柯西对$(X, \tilde{X})$，存在$V$的$X$ -特征基$\{v_i\}_{i \in \mathfrak{X}}$和$V$的$\tilde{X}$ -特征基$\{w_i\}_{i \in \mathfrak{X}}$，使得从$\{v_i\}_{i \in \mathfrak{X}}$到$\{w_i\}_{i \in \mathfrak{X}}$的转移矩阵是柯西的.我们表明，每一个柯西矩阵出现作为一个柯西对的过渡矩阵，以这种方式。给出了$V$上柯西对的等价类集与$\text{Mat}_\mathfrak{X}(\mathbb{K})$上柯西矩阵的置换等价类集之间的一个双射.
英文摘要：
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 $\text{Mat}_\mathfrak{X}(\mathbb{K})$.
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