Vandermonde Matrix Identity for Hilbert Matrix

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Theorem

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

Let:

\(\displaystyle H\) \(=\) \(\displaystyle \paren {\begin{smallmatrix}\displaystyle 1 & \dfrac {1} {2} & \cdots & \dfrac {1} {n} \\ \dfrac {1} {2} & \dfrac 1 {3} & \cdots & \dfrac {1} {n+1} \\ \vdots & \vdots & \cdots & \vdots \\ \dfrac {1} {n} & \dfrac {1} {n+1} & \cdots & \dfrac {1} {2n-1} \\ \end{smallmatrix} }\) Hilbert matrix of order $n$

Then:

\(\displaystyle H\) \(=\) \(\displaystyle -P V_x^{-1} V_y Q^{-1}\) Vandermonde Matrix Identity for Cauchy Matrix and Hilbert Matrix is Cauchy Matrix

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.

$\blacksquare$


Examples

$3 \times 3$ Matrix

Define polynomial root sets $\set {1,2,3}$ and $\set { 0,-1,-2}$ for Definition:Cauchy Matrix because Hilbert Matrix is Cauchy Matrix.

Illustrate $3\times 3$ case for Vandermonde Matrix Identity for Hilbert Matrix and value of Hilbert matrix determinant:

\(\displaystyle H\) \(=\) \(\displaystyle {\begin{pmatrix} \frac 1 {1} & \frac 1 {2} & \frac 1 {3} \\ \frac 1 {2} & \frac 1 {3} & \frac 1 {4} \\ \frac 1 {3} & \frac 1 {4} & \frac 1 {5} \\ \end{pmatrix} }\) Hilbert matrix of order $3$

Then:

\(\displaystyle H\) \(=\) \(\displaystyle -P V_x^{-1} V_y Q^{-1}\) Vandermonde Matrix Identity for Hilbert Matrix
\(\displaystyle \det \paren H\) \(=\) \(\displaystyle \dfrac {1} {2140}\) Determinant Product Theorem


Also see


Sources