Saturday, 12 July 2014

linear algebra - Finding the closed of form of the determinant of the Hilbert matrix

In my studies of matrix theory I came across the famous Hilbert matrix, which is a square n×n matrix H with entries given by: hij=1i+j1 and this is an example of a Cauchy matrix, which is a matrix Cn of the form cij=1xi+yj and for this matrix there is the well known formula for the determinant:



det(C)=1i<jn(xjxi)(yjyi)1i,jn(xi+yj)



Now I think I can substitute the sequences for the Hilbert matrix but I cannot see how to get the closed form they got here (under Properties):



det(H)=c4nc2n where cn=n1i=1in1=n1i=1i!




and I was hoping someone would please help me obtain the closed form. Thanks.

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