# Cholesky Decomposition of the Hilbert Matrix

I would like to have an analytical expression for the Cholesky decomposition of the following matrix: $$\mathbf A = \left [ \begin{array}{cccc} 1/1 & 1/2 & 1/3 & 1/4 \\ 1/2 & 1/3 & 1/4 & 1/5 \\ 1/3 & 1/4 & 1/5 & 1/6 \\ 1/4 & 1/5 & 1/6 & 1/7 \\ \end{array} \right ].$$

Really, I would be interested in any proof that it is positive definite.

• Do you mean, something like using WA? – Moo Jan 30 '16 at 4:27
• Thank you for your help, Moo. WA's output is helpful, but what I would really like is to be able to express the (i,j)th element of A in terms of i and j. – Garrett Jan 30 '16 at 4:30
• I would think searching for Cholesky Decomposition of Hilbert Matrix would locate something. – Moo Jan 30 '16 at 4:34
• Ah, it's called the "Hilbert Matrix"! Thanks, Moo! That's the lead I needed! – Garrett Jan 30 '16 at 4:41
• link.springer.com/article/10.1007%2FBF03167904#page-1 – Moo Jan 30 '16 at 4:58