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.
Also see
- Results about Cauchy matrices can be found here.
Source of Name
This entry was named for Augustin Louis Cauchy.
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: Exercises -- Second Set