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 $H$ 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.
$\blacksquare$
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 | |||||||||||
| \(\ds \map \det H\) | \(=\) | \(\ds \dfrac 1 {2140}\) | Determinant of Matrix Product |
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)