Last edited by JaneFairfax (2009-03-28 00:31:32)
Quite fascinating Jane. That's the square root of the discriminant for a polynomial with roots a1, ..., an. Nice find.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Yes, its a nice find.
There are a couple of proofs of the result on the French Wikipedia website: http://fr.wikipedia.org/wiki/Matrice_de_Vandermonde. (Note that the website uses the transpose of the matrix I posted instead.)
The result is easily verified for small values of n. Suppose the result is true for some natural number n and consider the (n+1)×(n+1) matrix.
By multiplying the ith row byand subtracting from the (i+1)th row we have
By the inductive hypothesis, the determinant of the smaller matrix is.
That completes the proof by induction.
Last edited by JaneFairfax (2009-03-28 02:33:49)