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 $a_{ij} = \dfrac 1 {x_i + y_j}$;
• or all of whose elements are in the form $a_{ij} = \dfrac 1 {x_i - y_j}$.

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 definitely 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:

$\begin{bmatrix} \dfrac 1 {x_1 - y_1} & \dfrac 1 {x_1 - y_2 } & \cdots & \dfrac 1 {x_1 - y_n} \\ \dfrac 1 {x_2 - y_1} & \dfrac 1 {x_2 - y_2 } & \cdots & \dfrac 1 {x_2 - y_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac 1 {x_m - y_1} & \dfrac 1 {x_m - y_2 } & \cdots & \dfrac 1 {x_m - y_n} \\ \end{bmatrix}$

Source of Name

This entry was named for Augustin Louis Cauchy.

Sources

• Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.3$