Category:Examples of Vandermonde Matrix Identity

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This category contains examples of use of Vandermonde Matrix Identity for Cauchy Matrix.

Assume values $\set { x_1,\ldots,x_n,y_1,\ldots,y_n }$ are distinct in matrix

\(\displaystyle C\) \(=\) \(\displaystyle \paren {\begin{smallmatrix} \dfrac {1} {x_1 - y_1} & \dfrac {1} {x_1 - y_2} & \cdots & \dfrac {1} {x_1 - y_n} \\ \dfrac {1} {x_2 - y_1} & \dfrac 1 {x_2 - y_2} & \cdots & \dfrac {1} {x_2 - y_n} \\ \vdots & \vdots & \cdots & \vdots \\ \dfrac {1} {x_n - y_1} & \dfrac {1} {x_n - y_2} & \cdots & \dfrac {1} {x_n - y_n} \\ \end{smallmatrix} }\) Cauchy matrix of order $n$

Then:

\(\displaystyle C\) \(=\) \(\displaystyle -P V_x^{-1} V_y Q^{-1}\) Vandermonde matrix identity for a Cauchy matrix