proof of determinant of the Vandermonde matrix

To begin, note that the determinantMathworldPlanetmath of the n×n Vandermonde matrixMathworldPlanetmath (which we shall denote as ‘Δ’) is a homogeneous polynomialMathworldPlanetmathPlanetmath of order n(n-1)/2 because every term in the determinant is, up to sign, the product of a zeroth power of some variable times the first power of some other variable , , the n-1-st power of some variable and 0+1++(n-1)=n(n-1)/2.

Next, note that if ai=aj with ij, then Δ=0 because two columns of the matrix would be equal. Since Δ is a polynomialMathworldPlanetmathPlanetmathPlanetmath, this implies that ai-aj is a factor of Δ. Hence,


where C is some polynomial. However, since both Δ and the product on the right hand side have the same degree, C must have degree zero, i.e. C must be a constant. So all that remains is the determine the value of this constant.

One way to determine this constant is to look at the coefficient of the leading diagonal, n(an)n-1. Since it equals 1 in both the determinant and the product, we conclude that C=1, hence

Title proof of determinant of the Vandermonde matrix
Canonical name ProofOfDeterminantOfTheVandermondeMatrix
Date of creation 2013-03-22 15:44:50
Last modified on 2013-03-22 15:44:50
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 10
Author rspuzio (6075)
Entry type Proof
Classification msc 15A57
Classification msc 65F99
Classification msc 65T50