Vandermonde matrix and determinant

I failed to mention the Vandermonde matrix because I couldn't see how it fit in with the story. Let x1, ..., xn be distinct numbers and let

P(x)= (x - x1)· ... ·(x - xn)
 = xn + a1xn-1 + a2xn-2 + ... + an.

Let

G =
|010......|
|001......|
|.....|
|0...1|
|-an-an-1-an-2...-a1|

be the companion matrix of P(x). Then (1, xi, xi2, ..., xin-1)T is an eigenvector of G with eigenvalue xi. Since the xi are distinct, these eigenvectors are linearly independent. Hence the Vandermonde matrix, whose columns are these vectors, is nonsingular.

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