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