\begin{definition}[Definition:Cauchy Matrix]
The '''Cauchy matrix''', commonly denoted $C_n$, can be found defined in two forms.
The '''Cauchy matrix''' is an $m \times n$ matrix whose elements are in the form:
: either $a_{ij} = \dfrac 1 {x_i + y_j}$
: or $a_{ij} = \dfrac 1 {x_i - y_j}$.
where $x_1, x_2, \ldots, x_m$ and $y_1, y_2, \ldots, y_n$ are elements of a field $F$.
They are of course equivalent, by taking $y'_j = -y_j$.
Note that $x_i + y_j$ (or $x_i - y_j$, depending on how the matrix is defined) must '''not''' be zero, or the element will be undefined.
Thus, writing the matrix out in full:
:$C_n := \begin{bmatrix}
\dfrac 1 {x_1 + y_1} & \dfrac 1 {x_1 + y_2 } & \cdots & \dfrac 1 {x_1 + y_n} \\
\dfrac 1 {x_2 + y_1} & \dfrac 1 {x_2 + y_2 } & \cdots & \dfrac 1 {x_2 + y_n} \\
\vdots & \vdots & \ddots & \vdots \\
\dfrac 1 {x_m + y_1} & \dfrac 1 {x_m + y_2 } & \cdots & \dfrac 1 {x_m + y_n} \\
\end{bmatrix}$
or:
:$C_n := \begin{bmatrix}
\dfrac 1 {x_1 - y_1} & \dfrac 1 {x_1 - y_2 } & \cdots & \dfrac 1 {x_1 - y_n} \\
\dfrac 1 {x_2 - y_1} & \dfrac 1 {x_2 - y_2 } & \cdots & \dfrac 1 {x_2 - y_n} \\
\vdots & \vdots & \ddots & \vdots \\
\dfrac 1 {x_m - y_1} & \dfrac 1 {x_m - y_2 } & \cdots & \dfrac 1 {x_m - y_n} \\
\end{bmatrix}$
\end{definition}