Hilbert matrices are well known to be ill-conditioned, with the columns being almost linearly dependent. On the wikipedia page, they state that the condition number grows as $$O((1+\sqrt{2})^{4n}/\sqrt{n}$$ I think the corresponding reference for this statement is the paper "The Condition Number of the Finite Segment of the Hilbert Matrix", by John Todd, but I couldn't manage to find a copy of it. I would really like to know what kind of tools one can use to proof a result like this one. Does anyone know this proof or at least can give a sketch ?
Since this proof might be involved, I'm also interested in any proof of a weaker lower bound that one can prove using "basic" tools.
Thanks!