Definition:Infinite Hilbert Matrix
Jump to navigation
Jump to search
Definition
The infinite Hilbert matrix is the infinite matrix whose elements are defined as:
- $a_{ij} = \dfrac 1 {i + j - 1}$
In full, it appears as:
- $\begin{bmatrix} 1 & \tfrac 1 2 & \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \ldots \\ \tfrac 1 2 & \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \ldots \\ \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \ldots \\ \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \ldots \\ \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \tfrac 1 {10} & \ldots \\ \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \tfrac 1 {10} & \tfrac 1 {11} & \ldots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix}$
Source of Name
This entry was named for David Hilbert.
Sources
- 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 $45$