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Deriving Inverse of Hilbert Matrix
The inverse of the Hilbert Matrix is made up entirely of integer entries, but I can't seem to find an elementary proof for that though, any hints?
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2$\begingroup$ The Wikipedia article gives an explicit, and not terribly complicated, expression for the entries of the inverse. Can't you just multiply the two matrices together to get a proof? $\endgroup$– Emil JeřábekCommented Mar 29, 2011 at 18:15
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3$\begingroup$ This came up before: mathoverflow.net/questions/47561/… $\endgroup$– FaisalCommented Mar 29, 2011 at 19:06
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2 Answers
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You can look at M.-D. Choi's paper
http://www.jstor.org/stable/pdfplus/2975779.pdf
(American Math Monthly, 1983, "Tricks or Treats with the Hilbert Matrix") for this, and much more.
answered Mar 29, 2011 at 19:09
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2$\begingroup$ Freely avaliable online: vigo.ime.unicamp.br/HilbertMatrix.pdf $\endgroup$ Commented Mar 29, 2011 at 19:20
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$\begingroup$ The solution provided in the paper uses the concept of the determinant, but this question is provided as an exercise problem in Hoffman and Kunze's book way before the determinant is introduced. Before presenting the determinant solution, the above paper refers to the direct computation as "cumbersome". $\endgroup$– fmardiniCommented Mar 29, 2011 at 22:09
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Look at the paper below for what you are asking and much more! There is a lot known about the Hilbert matrix.
Tricks or Treats with the Hilbert Matrix by Man-Duen Choi published at The American Mathematical Monthly, Vol. 90, No. 5 (May, 1983), pp. 301-312
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2$\begingroup$ Freely avaliable online: vigo.ime.unicamp.br/HilbertMatrix.pdf $\endgroup$ Commented Mar 29, 2011 at 19:18
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2$\begingroup$ In addition this two papers might be useful: ams.org/journals/proc/1958-009-01/S0002-9939-1958-0094626-0/… ams.org/journals/proc/1958-009-04/S0002-9939-1958-0099599-2/… $\endgroup$– ghtCommented Mar 29, 2011 at 19:19