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, ..., x_n be distinct numbers and let

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

Let

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

be the companion matrix of P(x). Then (1, x_i, x_i², ..., x_i^n-1)^T is an eigenvector of G with eigenvalue x_i. Since the x_i are distinct, these eigenvectors are linearly independent. Hence the Vandermonde matrix, whose columns are these vectors, is nonsingular.

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