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.

$\blacksquare$

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 $45$