Definition:Hilbert Matrix

Definition
A Hilbert matrix is an order $n$ square submatrix of the infinite Hilbert matrix, consists 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:
 * $$a_{ij} = \frac 1 {i + j - 1}$$

The order $6$ Hilbert matrix is:
 * $$\begin{bmatrix}

1 & \tfrac 1 2 & \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 \\[4pt] \tfrac 1 2 & \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 \\[4pt] \tfrac 1 3 & \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 \\[4pt] \tfrac 1 4 & \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 \\[4pt] \tfrac 1 5 & \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \tfrac 1 {10} \\[4pt] \tfrac 1 6 & \tfrac 1 7 & \tfrac 1 8 & \tfrac 1 9 & \tfrac 1 {10} & \tfrac 1 {11} \end{bmatrix}$$