Definition:Cauchy Matrix

Definition
The Cauchy matrix can be found defined in two forms.

The Cauchy matrix is an $$m \times n$$ matrix whose elements are in the form:
 * Either all of whose elements are in the form $$a_{ij} = \frac 1 {x_i + y_j}$$;
 * or all of whose elements are in the form $$a_{ij} = \frac 1 {x_i - y_j}$$.

where $$x_1, x_2, \ldots, x_m$$ and $$y_1, y_2, \ldots, y_n$$ be elements of a field $$F$$.

They are of course equivalent, by taking $$y'_j = -y_j$$.

Some sources insist that:
 * the elements $$x_1, x_2, \ldots, x_m$$ are all distinct;
 * the elements $$y_1, y_2, \ldots, y_n$$ are also all distinct.

If this is not the case, then its determinant is undefined.

Note that $$x_i + y_j$$ (or $$x_i - y_j$$, depending on how the matrix is defined) may not be zero, or the element will be undefined.

Thus, writing the matrix out in full, we get:
 * $$\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:
 * $$\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}$$