Vandermonde matrix - Knowlpedia

For scalars $x_1,\ldots,x_m$ , the Vandermonde matrix is

V_{ij}=x_j^{i-1},\qquad 1\le i,j\le m.

Its determinant is $\prod_{1\le j<k\le m}(x_k-x_j)$ , so it is invertible precisely when the nodes are distinct.

Determinant argument

The determinant is an alternating polynomial in the nodes, hence divisible by every difference $x_k-x_j$ . Its total degree equals that of their product, so the quotient is constant. Comparing the coefficient of $x_2x_3^2\cdots x_m^{m-1}$ gives constant one. Transposing the matrix changes the row-column convention but preserves the determinant.