The Parameswaran argument is summarised at http://en.wikipedia.org/wiki/Pfaffian WFL On 10/23/08, Dan Asimov <dasimov@earthlink.net> wrote:
Gene asked:
<< 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?
Here's a cite for what appears to be an elementary 1-page proof:
Skew-Symmetric Determinants Author(s): S. Parameswaran Source: The American Mathematical Monthly, Vol. 61, No. 2 (Feb., 1954), p. 116
I believe there is a more informative but more advanced proof in "Characteristic Classes" by J. Milnor (Princeton orange series).
--Dan
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun