2
$\begingroup$

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.

  • 0
    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'.2012-12-29

1 Answers 1