PlanetMath: Cauchy matrix

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Cauchy matrix

(Definition)

Let , $x_2,\ldots, x_m$ , and , $y_2 \ldots, y_n$ be elements in a field , satisfying the properties that

$x_1, \ldots, x_m$ are distinct,
$y_1, \ldots, y_n$ are distinct, and
$x_i+y_j\neq 0$ for $1\leq i \leq m$ , $1\leq j \leq n$ .

The matrix

$\displaystyle \begin{bmatrix} \frac{1}{x_1+y_1} & \frac{1}{x_1+y_2} & \cdots &\... ...\frac{1}{x_m+y_1} & \frac{1}{x_m+y_2} & \cdots &\frac{1}{x_m+y_n} \end{bmatrix}$

is called a Cauchy matrix over

The determinant of a square Cauchy matrix is

$\displaystyle \frac{ \prod_{i<j} (x_i-x_j)(y_i-y_j) } {\prod_{ij} (x_i+y_j)}$

Since 's are distinct and 's are distinct by definition, a square Cauchy matrix is non-singular. Any submatrix of a rectangular Cauchy matrix has full rank.

"Cauchy matrix" is owned by kshum.

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Also defines:

Cauchy matrices

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Cross-references: rank, submatrix, non-singular, square, determinant, matrix, field
There is 1 reference to this entry.

This is version 6 of Cauchy matrix, born on 2004-07-30, modified 2006-03-15.
Object id is 6050, canonical name is CauchyMatrix.
Accessed 4210 times total.

Classification:

AMS MSC:

15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices )

Pending Errata and Addenda

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