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[parent] proof of determinant of the Vandermonde matrix (Proof)

To begin, note that the determinant of the $ n \times n$ Vandermonde matrix (which we shall denote as `$ \Delta$') is a homogeneous polynomial 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 , $ \ldots$, the $ n-1$-st power of some variable and $ 0 + 1 + \cdots + (n-1) = n(n-1)/2$.

Next, note that if $ a_i = a_j$ with $ i \neq j$, then $ \Delta = 0$ because two columns of the matrix would be equal. Since $ \Delta$ is a polynomial, this implies that $ a_i - a_j$ is a factor of $ \Delta$. Hence,

$\displaystyle \Delta = C \prod_{1 \leq i < j \leq n}(a_j - a_i) $
where C is some polynomial. However, since both $ \Delta$ 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, $ \prod_n (a_n)^{n-1}$. Since it equals 1 in both the determinant and the product, we conclude that $ C = 1$, hence

$\displaystyle \Delta = \prod_{1 \leq i < j \leq n}(a_j - a_i). $



"proof of determinant of the Vandermonde matrix" is owned by rspuzio. [ full author list (2) ]
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Cross-references: diagonal, coefficient, degree, right hand side, factor, implies, polynomial, matrix, columns, variable, power, product, term, order, homogeneous polynomial, Vandermonde matrix, determinant

This is version 7 of proof of determinant of the Vandermonde matrix, born on 2006-03-08, modified 2006-11-03.
Object id is 7699, canonical name is PrrofOfDeterminantOfTheVandermondeMatrix.
Accessed 6168 times total.

Classification:
AMS MSC: 65T50 (Numerical analysis :: Numerical methods in Fourier analysis :: Discrete and fast Fourier transforms)
 65F99 (Numerical analysis :: Numerical linear algebra :: Miscellaneous)
 15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices )
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