\begin{definition}[Definition:Infinite Hilbert Matrix]
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:
:$\begin{bmatrix}
1 & \tfrac 1 2 & \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \ldots \\
\tfrac 1 2 & \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \ldots \\
\tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \ldots \\
\tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \ldots \\
\tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \tfrac 1 {10} & \ldots \\
\tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \tfrac 1 {10} & \tfrac 1 {11} & \ldots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots 
\end{bmatrix}$
{{NamedforDef|David Hilbert|cat = Hilbert}}
\end{definition}