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:
| \(\ds H\) | \(=\) | \(\ds \paren {\begin{smallmatrix}\displaystyle 1 & \dfrac {1} {2} & \cdots & \dfrac {1} {n} \\ \dfrac {1} {2} & \dfrac 1 {3} & \cdots & \dfrac {1} {n+1} \\ \vdots & \vdots & \cdots & \vdots \\ \dfrac {1} {n} & \dfrac {1} {n+1} & \cdots & \dfrac {1} {2n-1} \\ \end{smallmatrix} }\) | Hilbert matrix of order $n$ |
Then:
| \(\ds H\) | \(=\) | \(\ds -P V_x^{-1} V_y Q^{-1}\) | 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:
| \(\ds H\) | \(=\) | \(\ds {\begin{pmatrix} \frac 1 {1} & \frac 1 {2} & \frac 1 {3} \\ \frac 1 {2} & \frac 1 {3} & \frac 1 {4} \\ \frac 1 {3} & \frac 1 {4} & \frac 1 {5} \\ \end{pmatrix} }\) | Hilbert matrix of order $3$ |
Then:
| \(\ds H\) | \(=\) | \(\ds -P V_x^{-1} V_y Q^{-1}\) | Vandermonde Matrix Identity for Hilbert Matrix | |||||||||||
| \(\ds \det \paren H\) | \(=\) | \(\ds \dfrac {1} {2140}\) | Determinant Product Theorem |
Also see
Sources
- 1944: A.C. Aitken: Determinants and Matrices (3rd ed.): Chapter $\text{VI}$. $47$: Alternant Matrices and Determinants
- March 1992: Roderick Gow: Cauchy's matrix, the Vandermonde matrix and polynomial interpolation (Bull. Irish Math. Soc. Vol. 28: pp. 45 – 52)