Vandermonde Matrix Identity for Cauchy Matrix

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

Then:

Definitions of Vandermonde matrices $V_x$, $V_y$ and diagonal matrices $P$, $Q$:


 * $\displaystyle V_x=\paren {\begin{smallmatrix}

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{smallmatrix} },\quad V_y=\paren {\begin{smallmatrix} 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{smallmatrix} }$ Vandermonde matrices


 * $\displaystyle P= \paren {\begin{smallmatrix}

p_1(x_1) & \cdots & 0 \\ \vdots  & \ddots  & \vdots \\ 0       & \cdots  & p_n(x_n) \\ \end{smallmatrix} }, \quad Q= \paren {\begin{smallmatrix} p(y_1) & \cdots  & 0 \\ \vdots & \ddots  & \vdots \\ 0      & \cdots  & p(y_n) \\ \end{smallmatrix} }$ Diagonal matrices

Definitions of polynomials $p$, $p_1$, $\ldots$, $p_n$:


 * $\displaystyle p(x) = \prod_{i \mathop = 1}^n \paren {x - x_i}$


 * $\displaystyle p_k(x) = \dfrac{ \map p x}{x-x_k} = \prod_{i \mathop = 1,i \mathop \ne k}^n \, \paren {x - x_i}$, $1 \mathop \le k \mathop \le n$

Proof
Matrices $P$ and $Q$ are invertible because all diagonal elements are nonzero.

For $1\le i \le n$ express polynomial $p_i$ as:


 * $\displaystyle \map {p_i} {x} = \sum_{k \mathop = 1}^n a_{ik} x^{k-1}$

Then:

Use second equation $\map {p_i} {y_j} = \dfrac{ \map {p} {y_j} }{y_j - x_i}$:



Also see
Vandermonde Matrix Identity for Cauchy Matrix/Example 3x3

Vandermonde Determinant

Inverse of Vandermonde Matrix

Hilbert Matrix is Cauchy Matrix

Inverse of Cauchy Matrix

Inverse of Hilbert Matrix

Value of Cauchy Determinant

Sum of Elements in Inverse of Cauchy Matrix

Sum of Elements in Inverse of Hilbert Matrix

Historical Note
established (1992) the Theorem using interpolation polynomials and change of basis facts. The Vandermonde matrices had ones along the first row. Aitken (1944) defines alternant, Vandermonde matrices a special case. Terms bialternant and double alternant apply to Cauchy matrices.