Theorem: Value of Cauchy Determinant

Let $D_n$ be a Cauchy determinant of order $n$:
:<nowiki>$\begin{vmatrix}
\dfrac 1 {x_1 + y_1} & \dfrac 1 {x_1 + y_2} & \cdots & \dfrac 1 {x_1 + y_n} \\
\dfrac 1 {x_2 + y_1} & \dfrac 1 {x_2 + y_2} & \cdots & \dfrac 1 {x_2 + y_n} \\
\vdots & \vdots & \ddots & \vdots \\
\dfrac 1 {x_n + y_1} & \dfrac 1 {x_n + y_2} & \cdots & \dfrac 1 {x_n + y_n} \\
\end{vmatrix}$</nowiki>
Then the value of $D_n$ is given by:
:$D_n = \dfrac {\ds \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_j - x_i} \paren {y_j - y_i} } {\ds \prod_{1 \mathop \le i, \, j \mathop \le n} \paren {x_i + y_j} }$
Let $D_n$ be given by:
:<nowiki>$\begin {vmatrix}
\dfrac 1 {x_1 - y_1} & \dfrac 1 {x_1 - y_2} & \cdots & \dfrac 1 {x_1 - y_n} \\
\dfrac 1 {x_2 - y_1} & \dfrac 1 {x_2 - y_2} & \cdots & \dfrac 1 {x_2 - y_n} \\
\vdots & \vdots & \ddots & \vdots \\
\dfrac 1 {x_n - y_1} & \dfrac 1 {x_n - y_2} & \cdots & \dfrac 1 {x_n - y_n} \\
\end {vmatrix}$</nowiki>
Then its determinant is given by:
:$D_n = \dfrac {\ds \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_j - x_i} \paren {y_i - y_j} } {\ds \prod_{1 \mathop \le i, \, j \mathop \le n} \paren {x_i - y_j} }$

Proof:
Let:
{{begin-eqn}}
{{eqn | l = C
      | r = <nowiki>\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_n - y_1} & \dfrac 1 {x_n - y_2} & \cdots & \dfrac 1 {x_n - y_n} \\
\end {bmatrix}</nowiki>
}}
{{end-eqn}}
To be proved:
{{begin-eqn}}
{{eqn | l = \map \det C
      | r = \dfrac {\ds \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_j - x_i} \paren {y_i - y_j} } {\ds \prod_{1 \mathop \le i, j \mathop \le n} \paren {x_i - y_j} }
      | c = replacing $y_k \to -y_k$ in $C$ and $\map \det C$
}}
{{end-eqn}}
Assume hereafter that set $\set {x_1, \ldots, x_n, y_1, \ldots, y_n}$ consists of distinct values, because otherwise $\map \det C$ is undefined or zero.
'''Preliminaries''':
Vandermonde Matrix Identity for Cauchy Matrix supplies matrix equation
:$(1): \quad - C = P V_x^{-1} V_y Q^{-1}$
:Definitions of symbols:
::<nowiki>$V_x = \begin {pmatrix}
1         & 1         & \cdots & 1 \\
x_1       & x_2       & \cdots & x_n \\
\vdots    & \vdots    & \ddots & \vdots \\
{x_1}^{n - 1} & {x_2}^{n - 1} & \cdots & {x_n}^{n - 1} \\
\end {pmatrix}$</nowiki>
::<nowiki>$V_y = \begin {pmatrix}
1         & 1         & \cdots & 1 \\
y_1       & y_2       & \cdots & y_n \\
\vdots    & \vdots    & \ddots & \vdots \\
{y_1}^{n - 1} & {y_2}^{n - 1} & \cdots & {y_n}^{n - 1} \\
\end {pmatrix}$</nowiki> {{Defof|Vandermonde Matrix|subdef = Formulation 1}}
::<nowiki>$P = \begin {pmatrix}
\map {p_1} {x_1} &  \cdots & 0 \\
\vdots   & \ddots  & \vdots \\
0        & \cdots  & \map {p_n} {x_n} \\
\end {pmatrix}$</nowiki>
::<nowiki>$Q = \begin {pmatrix}
\map p {y_1}  & \cdots  & 0 \\
\vdots  & \ddots  & \vdots \\
0       & \cdots  & \map p {y_n)} \\
\end {pmatrix}$</nowiki> {{Defof|Diagonal Matrix}}
:$1 \mathop \le k \mathop \le n$ Polynomials
::$\ds \map p x  = \prod_{i \mathop = 1}^n \paren {x - x_i}$
::$\ds \map {p_k} x = \prod_{i \mathop = 1, i \mathop \ne k}^n \paren {x - x_i}$
'''Determinant of $C$ Calculation''':
{{begin-eqn}}
{{eqn | l = C
      | r = \paren {-I} P V_x^{-1} V_y Q^{-1}
      | c = Vandermonde Matrix Identity for Cauchy Matrix 
Symbol $I$ is the $n \times n$ identity matrix.
}}
{{eqn | n = 2
      | l = \map \det C
      | r = \map \det {-I} \map \det P \map \det {V_x^{-1} } \map \det {V_y} \map \det {Q^{-1} }
      | c = Determinant of Matrix Product
}}
{{eqn | r = \paren {-1}^n \map \det I \map \det P \map \det {V_x^{-1} } \map \det {V_y} \map  \det {Q^{-1} }
     | c = Effect of Elementary Row Operations on Determinant: factoring constant $-1$ from all rows of $-I$
}}
{{eqn | r = \paren {-1}^n \map \det I \dfrac {\map \det p \map \det {V_y} } {\map \det Q \map \det {V_x} }
      | c = Matrix Product with Adjugate Matrix and Determinant of Matrix Product
}}
{{end-eqn}}
'''Lemma''': $\map \det P = \paren {-1}^m \paren {\map \det {V_x} }^2$ where $m = \frac 1 2 n \paren {n - 1}$
:'''Details''': Determinant $\map \det P$ expands to:
{{begin-eqn}}
{{eqn | l = \prod_{j \mathop = 1}^n \map {p_j} {x_j} 
      | r = \prod_{j \mathop = 1}^n \prod_{k \mathop = 1, k \mathop \ne j}^n \paren {x_j - x_k}
      | c = Definition of polynomials $\map {p_j} x$
}}
{{end-eqn}}
:Pair factors $\paren {x_r - x_s}$ and $\paren {x_s - x_r}$ into factor $-\paren {x_s - x_r}^2$, then:
{{begin-eqn}}
{{eqn | l = \map \det P
      | r = \paren {-1}^m \paren {\prod_{1 \mathop \le r \mathop < s \mathop \le n} \paren {x_s - x_r }^2}
      | c = where $m = 1 + \cdots + \paren {n - 1} = \dfrac 1 2 n \paren {n - 1}$
}}
{{eqn | r = \paren {-1}^m \paren {\map \det {V_x} }^2
      | c = Value of Vandermonde Determinant
}}
{{end-eqn}}
{{qed|lemma}}
Apply the '''Lemma''' to equation (2):
{{begin-eqn}}
{{eqn | l = \map \det C
      | r = \paren {-1}^{n + m} \paren {\dfrac {\map \det {V_x} \map \det {V_y} } {\map \det Q} }
      | c = $m = \dfrac 1 2 n \paren {n - 1}$
}}
{{eqn | r = \paren {-1}^{n + m} \dfrac {\ds \prod_{1 \mathop \le m \mathop < k \mathop \le n}^{\phantom n} \paren {x_k - x_m} \prod_{1 \mathop \le m \mathop < k \mathop \le n} \paren {y_k - y_m} } {\ds \prod_{k \mathop = 1}^n \map p {y_k} }
      | c = Value of Vandermonde Determinant and Determinant of Diagonal Matrix
}}
{{eqn | r = \paren {-1}^{n + m} \dfrac {\ds \prod_{1 \mathop \le m \mathop < k \mathop \le n}^{\phantom n} \paren {x_k - x_m} \paren {y_k - y_m} } {\ds \prod_{k \mathop = 1}^n \prod_{j \mathop = 1}^n \paren {y_k - x_j} }
      | c = Definition of $\map p x$
}}
{{eqn | r = \paren {-1}^{n + m} \dfrac {\ds \paren {-1}^m \prod_{1 \mathop \le m \mathop < k \mathop \le n}^{\phantom n} \paren {x_k - x_m} \paren {y_m - y_k} } {\ds \paren {-1}^{n^2} \prod_{k \mathop = 1}^n \prod_{j \mathop = 1}^n \paren {x_j - y_k} }
      | c = factoring out changed signs
}}
{{eqn | r = \dfrac {\ds \prod_{1 \mathop \le m \mathop < k \mathop \le n}^{\phantom n} \paren {x_k - x_m} \paren {y_m - y_k} } {\ds \prod_{k \mathop = 1}^n \prod_{j \mathop = 1}^n \paren {x_j - y_k} }
      | c = as all signs cancel
}}
{{end-eqn}}
{{qed}}