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Is it possible to express the product: $$ \frac{\prod_{i < j} (a_i - a_j)(b_i - b_j) }{\prod_{i,j} (a_i - b_j) }$$ as the determinant of a single matrix ?

This comes from a physics paper. Should be similar to a Vandermonde determinant.

EDIT: Obviously, I do not want a $1 \times 1$ matrix whose single entry as the answer. Why are Hilbert matrices or Toeplitz matrices or Cauchy matrices, the natural choices then ? Sorry.

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note that this still doesn't fix it, since I could still construct a diagonal matrix with say $a_1 - a_2$ as the first entry, $a_2 - a_3$ as the second entry, and the rest of the product as the third entry, and the rest of the entries as 1. Of course, we could also have a diagonal matrix where the entires are these 'polynomials'. –  Calvin Lin Dec 29 '12 at 23:50

1 Answer 1

Perhaps you should be specific about what you want? A diagonal matrix with that value in the first entry and 1 everywhere else (on the diagonal) will suffice.

You can't request that the coefficients be polynomials, since your determinant has 'degree' -n.

You might want to check out Hilbert matrices, which have entries $\frac {1}{a_i + b_j}$ and determinant $\frac { \prod (a_i - a_j)(b_i-b_j)}{\prod (a_i + b_j) }$.

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@Ofir, the hope was that he would realize that. –  Calvin Lin Dec 29 '12 at 14:32
    
Replace all the $b_i$ by $-b_i$ in the Hilbert matrix and you'll get exactly what the OP wants (up to $\pm 1$). –  David Speyer Dec 29 '12 at 18:10
    
@DavidSpeyer Yes, the hope is that OP would realize that, which was my reply to Ofir, who then pulled his comment. –  Calvin Lin Dec 29 '12 at 19:03
    
David is correct, but I have to think about how to fix my wording. –  john mangual Dec 29 '12 at 23:44

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