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+j−1 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)=∏1≤i<j≤n(xj−xi)(yj−yi)∏1≤i,j≤n(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=∏n−1i=1in−1=∏n−1i=1i!
and I was hoping someone would please help me obtain the closed form. Thanks.
No comments:
Post a Comment