Definition:Cauchy Matrix

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Definition

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:

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


Also defined as

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.


Also known as

Some sources report this as Cauchy's matrix.


Source of Name

This entry was named for Augustin Louis Cauchy.


Sources