It's not strictly relevant, but I didn't know about it, and maybe nor did other people: an algorithm for evaluating determinants and Pfaffians without division in O(n^4) operations, which claims to outperform elimination for quite small order n for matrices with rational polynomial entries. [There is included the proof requested: short maybe, but elementary it ain't.] The URL is http://page.mi.fu-berlin.de/rote/Papers/abstract/Division-free+algorithms+fo... WFL On 10/23/08, Eugene Salamin <gene_salamin@yahoo.com> wrote:
The determinant of a matrix is a polynomial in its elements. For an antisymmetric matrix of odd order, the determinant is zero, while for an antisymmetric matrix of even order, this polynomial factors into the square of another polynomial, called the Pfaffian. Does anyone know of a simple, easy to follow, proof that the determinant factors as asserted?
Gene
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