Definition:Hilbert Matrix
Definition
A Hilbert matrix is an order $n$ square submatrix of the infinite Hilbert matrix, consisting of the elements in the first $n$ rows and columns of that matrix.
Thus it is an $n \times n$ matrix whose elements are defined as:
- $a_{i j} = \dfrac 1 {i + j - 1}$
The order $n$ Hilbert matrix is often denoted $H_n$.
Examples
$3 \times 3$ Hilbert Matrix
The order $3$ Hilbert matrix is:
- $H_3 = \begin {bmatrix}
1 & \tfrac 1 2 & \tfrac 1 3 \\ \tfrac 1 2 & \tfrac 1 3 & \tfrac 1 4 \\ \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 \\ \end {bmatrix}$
$6 \times 6$ Hilbert Matrix
The order $6$ Hilbert matrix is:
- $\begin {bmatrix}
1 & \tfrac 1 2 & \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 \\ \tfrac 1 2 & \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 \\ \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 \\ \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 \\ \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \tfrac 1 {10} \\ \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \tfrac 1 {10} & \tfrac 1 {11} \end {bmatrix}$
Also see
- Results about Hilbert matrices can be found here.
Source of Name
This entry was named for David Hilbert.
Historical Note
The concept of the Hilbert matrix was introduced by David Hilbert in $1894$.
Sources
- 1961: John Todd: Computational problems concerning the Hilbert matrix (J. Res. Natl. Bur. Stand. Ser. B Vol. 65: pp. 19 – 22)
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Hilbert matrix
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: Exercise $45$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Hilbert matrix
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hilbert matrix