Vandermonde Matrix Identity for Cauchy Matrix/Examples
Examples of Use of Vandermonde Matrix Identity for Cauchy Matrix
$3 \times 3$ Matrix
Illustrate $3 \times 3$ case for Vandermonde Matrix Identity for Cauchy Matrix and Value of Cauchy Determinant.
Let $C$ denote the Cauchy matrix of order $3$:
- $C = \begin {pmatrix}
\dfrac 1 {x_1 - y_1} & \dfrac 1 {x_1 - y_2} & \dfrac 1 {x_1 - y_3} \\ \dfrac 1 {x_2 - y_1} & \dfrac 1 {x_2 - y_2} & \dfrac 1 {x_2 - y_3} \\ \dfrac 1 {x_3 - y_1} & \dfrac 1 {x_3 - y_2} & \dfrac 1 {x_3 - y_3} \\ \end{pmatrix}$
where the values in $\set {x_1, x_2, x_3, y_1, y_2, y_3}$ are assumed to be distinct.
Then:
| \(\ds C\) | \(=\) | \(\ds -P V_x^{-1} V_y Q^{-1}\) | Vandermonde Matrix Identity for Cauchy Matrix | |||||||||||
| \(\ds \map \det C\) | \(=\) | \(\ds \paren {-1}^3 \dfrac {\paren {x_3 - x_1} \paren {x_3 - x_2} \paren {x_2 - x_1} \paren {y_3 - y_1} \paren {y_3 - y_2} \paren {y_2 - y_1} }
{\paren {x_1 - y_1} \paren {x_1 - y_2} \paren {x_1 - y_3} \paren {x_2 - y_1} \paren {x_2 - y_2} \paren {x_2 - y_3} \paren {x_3 - y_1} \paren {x_3 - y_2} \paren {x_3 - y_3} }\)
|
Determinant of Matrix Product |
$n \times n$ Matrix
The methods of the $3 \times 3$ example apply unchanged for the general $n \times n$ Cauchy matrix:
Assume values $\set {x_1, \ldots, x_n, y_1, \ldots, y_n}$ are distinct. Then:
- $\map \det {\begin{smallmatrix}
\dfrac 1 {x_1 - y_1} & \dfrac 1 {x_1 - y_2} & \cdots & \dfrac 1 {x_1 - y_n} \\ \dfrac 1 {x_2 - y_1} & \dfrac 1 {x_2 - y_2} & \cdots & \dfrac 1 {x_2 - y_n} \\ \vdots & \vdots & \cdots & \vdots \\ \dfrac 1 {x_n - y_1} & \dfrac 1 {x_n - y_2} & \cdots & \dfrac 1 {x_n - y_n} \\ \end{smallmatrix} } = \paren {-1}^n \dfrac {\ds \prod_{1 \mathop \le j \mathop < i \mathop \le n} \paren {x_i - x_j} \quad \prod_{1 \mathop \le j \mathop < i \mathop \le n} \paren {y_i - y_j} } {\ds \prod_{i \mathop = 1}^n \prod_{j \mathop = 1}^n \paren {x_i - y_j} }$ Value of Cauchy Determinant
Assume values $\set {x_1, \ldots, x_n, -y_1, \ldots, -y_n}$ are distinct, then replace in the preceding equation $y_i$ by $-y_i$, $1 \le i \le n$:
- $\map \det {\begin{smallmatrix}
\dfrac 1 {x_1 + y_1} & \dfrac 1 {x_1 + y_2} & \cdots & \dfrac 1 {x_1 + y_n} \\ \dfrac 1 {x_2 + y_1} & \dfrac 1 {x_2 + y_2} & \cdots & \dfrac 1 {x_2 + y_n} \\ \vdots & \vdots & \cdots & \vdots \\ \dfrac 1 {x_n + y_1} & \dfrac 1 {x_n + y_2} & \cdots & \dfrac 1 {x_n + y_n} \\ \end{smallmatrix} } = \paren {-1}^n \dfrac {\ds \prod_{1 \mathop \le j \mathop < i \mathop \le n} \paren {x_i - x_j} \quad \prod_{1 \mathop \le j \mathop < i \mathop \le n} \paren {y_j - y_i} } {\ds \prod_{i \mathop = 1}^n \prod_{j \mathop = 1}^n \paren {x_i + y_j} }$ Value of Cauchy Determinant
$\blacksquare$