Friday, 26 April 2013

linear algebra - Determinant of ntimesn matrix with parameter



Problem:





Let δR+ and nN. The matrix An=(ai,j)Rn×n is defined as



ai,j=i2k=0((j1)δ+nk)
Prove that
det




So there is




A_1 = \pmatrix{ 1\\ }



A_2 = \pmatrix{ 1&1\\ 2&\delta+2\\ }



A_3 = \pmatrix{ 1&1&1\\ 3&\delta+3&2\delta+3\\ 6&(\delta+3)(\delta+2)&(2\delta+3)(2\delta+2)\\ }



\vdots



I eventually managed to prove it by converting the matrix to upper triangular using elementary row operations, but the proof is just too complicated, involves things like (k−2)\delta+n−(i−(k−1))+1)-th multiples of certain rows (for that matter it is quite long, so I am not fully including it here). So it somehow feels like not the best possible way to do this.



What are some another ways to prove this?



Answer



Finally found this in a literature, namely Calculation of some determinants using the s-shifted factorial by Jean-Marie Normand. In Lemma 1 he proves equation (3.5):




For complex numbers z_j, s, b_i we have\tag{1}\det [(b_i+z_j)_{s;i}]_{i,j=0,\dots,n-1}=\prod_{0\leq i \leq j \leq n-1}(z_j-z_i) where
(z)_{s;i}=z(z+s)\dots(z+(n-1)s) is called s-shifted factorial and \prod_{0\leq i\leq j\leq n-1}(z_j-z_i) is a Vandermonde determinant.




Furthermore Appendix B shows in equation (B.5) that for special case b_i=0, z_j=b+aj one has





\det[(b_i+z_j)_{s;i}]_{i,j=0,\dots,n-1}=\det[(b+aj)_{s;i}]_{i,j=0,\dots,n-1}=a^{n(n-1)/2}\prod_{j=0}^{n-1}j!.\tag{2}




Before applying to our problem, we first need to re-index the range to i,j=0,\dots,n-1, then we have a_{i,j}=(j\delta+n)_{1;i}. Now choosing b_i=0, a=\delta, b=n, s=1 in (2) gives exactly the statement we wanted to prove.


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