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 $H$ be the Hilbert matrix of order $n$:


 * $H = \begin {pmatrix}

1 & \dfrac 1 2 & \cdots & \dfrac 1 n \\ \dfrac 1 2 & \dfrac 1 3  & \cdots & \dfrac 1 {n + 1} \\ \vdots    & \vdots       & \cdots & \vdots \\ \dfrac 1 n & \dfrac 1 {n + 1} & \cdots & \dfrac 1 {2 n - 1} \end {pmatrix}$

Then from Vandermonde Matrix Identity for Cauchy Matrix and Hilbert Matrix is Cauchy Matrix:
 * $H = -P V_x^{-1} V_y Q^{-1}$

where $V_x$, $V_y$ are Vandermonde matrices:


 * $V_x = \begin {pmatrix}

1        & 1         & \cdots & 1 \\ 1        & 2       & \cdots & n \\ \vdots   & \vdots    & \ddots & \vdots \\ 1 & 2^{n - 1} & \cdots & n^{n -1 } \\ \end {pmatrix}, \quad V_y = \begin {pmatrix} 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 {pmatrix}$

and $P$, $Q$ are diagonal matrices:


 * $P = \begin {pmatrix}

\map {p_1} 1 & \cdots & 0 \\ \vdots  & \ddots  & \vdots \\ 0       & \cdots  & \map {p_n} n \\ \end {pmatrix}, \quad Q = \begin {pmatrix} \map p 0 & \cdots & 0 \\ \vdots & \ddots  & \vdots \\ 0      & \cdots  & \map p {-n + 1} \\ \end {pmatrix}$

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


 * $\ds \map p x = \prod_{i \mathop = 1}^n \paren {x - i}$


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

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


 * Definition:Vandermonde Determinant


 * Inverse of Vandermonde Matrix


 * Inverse of Hilbert Matrix


 * Sum of Elements in Inverse of Hilbert Matrix