# Hilbert Matrix is Cauchy Matrix

## Theorem

A Hilbert matrix is a special case of a Cauchy matrix.

## Proof

By definition of Hilbert matrix, the element $a_{i j}$ is:

$a_{i j} = \dfrac 1 {i + j - 1}$

For all $i, j \in \Z$ such that $1 \le i \le n$ and $1 \le j \le n$, let:

$x_i = i$
$y_j = j - 1$

Then:

$a_{i j} = \dfrac 1 {x_i + y_j}$

The result follows by definition of a Cauchy matrix.

$\blacksquare$