Vandermonde Matrix Identity for Hilbert Matrix

Theorem
Define polynomial root sets $\set {1,2,\ldots, n}$ and $\set { 0,-1,\ldots,-n+1}$ for Definition:Cauchy Matrix.

Let:

Then:

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


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

1        & 1         & \cdots & 1 \\ 1      & 2       & \cdots & n \\ \vdots   & \vdots    & \ddots & \vdots \\ 1 & 2^{n-1} & \cdots & n^{n-1} \\ \end{smallmatrix} },\quad V_y=\paren {\begin{smallmatrix} \displaystyle 1        & 1         & \cdots & 1 \\ 0      & -1       & \cdots & -n+1 \\ \vdots   & \vdots    & \ddots & \vdots \\ 0 & \paren {-1}^{n-1} & \cdots & \paren {-n+1}^{n-1} \\ \end{smallmatrix} }$ Vandermonde matrices


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

p_1(1) & \cdots & 0 \\ \vdots  & \ddots  & \vdots \\ 0       & \cdots  & p_n(n) \\ \end{smallmatrix} }, \quad Q= \paren {\begin{smallmatrix} p(0) & \cdots  & 0 \\ \vdots & \ddots  & \vdots \\ 0      & \cdots  & p(-n+1) \\ \end{smallmatrix} }$ Diagonal matrices

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


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


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

Proof
Apply Vandermonde Matrix Identity for Cauchy Matrix and Hilbert Matrix is Cauchy Matrix.

Matrices $V_x$ and $V_y$ are invertible by Inverse of Vandermonde Matrix.

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

Also see

 * Hilbert Matrix is Cauchy Matrix


 * Vandermonde Determinant


 * Inverse of Vandermonde Matrix


 * Inverse of Hilbert Matrix


 * Sum of Elements in Inverse of Hilbert Matrix