Vandermonde Matrix Identity for Cauchy Matrix/Examples/nxn

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Example of Vandermonde Matrix Identity for Cauchy Matrix

The methods of the $3\times 3$ example apply unchanged for the general $n \times n$ Cauchy matrix:

Assume values $\left\{ x_1,\ldots,x_n,y_1,\ldots,y_n\right\}$ are distinct. Then:

$\det \paren {\begin{smallmatrix} \frac {1} {x_1 - y_1} & \frac {1} {x_1 - y_2} & \cdots & \frac {1} {x_1 - y_n} \\ \frac {1} {x_2 - y_1} & \frac 1 {x_2 - y_2} & \cdots & \frac {1} {x_2 - y_n} \\ \vdots & \vdots & \cdots & \vdots \\ \frac {1} {x_n - y_1} & \frac {1} {x_n - y_2} & \cdots & \frac {1} {x_n - y_n} \\ \end{smallmatrix} } = (-1)^n \dfrac {\prod_{1 \mathop \le j < i \mathop \le n} \paren {x_i - x_j} \quad \prod_{1 \mathop \le j \mathop < i \mathop \le n} \paren {y_i - y_j} } {\prod_{i \mathop = 1}^n \prod_{j \mathop = 1}^n \paren {x_i - y_j} }$ Value of Cauchy Determinant

Assume values $\left\{ x_1,\ldots,x_n,-y_1,\ldots,-y_n\right\}$ are distinct, then replace in the preceding equation $y_i$ by $-y_i$, $1\le i \le n$:

$\det \paren {\begin{smallmatrix} \frac {1} {x_1 + y_1} & \frac {1} {x_1 + y_2} & \cdots & \frac {1} {x_1 + y_n} \\ \frac {1} {x_2 + y_1} & \frac 1 {x_2 + y_2} & \cdots & \frac {1} {x_2 + y_n} \\ \vdots & \vdots & \cdots & \vdots \\ \frac {1} {x_n + y_1} & \frac {1} {x_n + y_2} & \cdots & \frac {1} {x_n + y_n} \\ \end{smallmatrix} } = (-1)^n \dfrac {\prod_{1 \mathop \le j \mathop < i \mathop \le n} \paren {x_i - x_j} \quad \prod_{1 \mathop \le j \mathop < i \mathop \le n} \paren {y_j - y_i} } {\prod_{i \mathop = 1}^n \prod_{j \mathop = 1}^n \paren {x_i + y_j} }$ Value of Cauchy Determinant

$\blacksquare$