Vandermonde determinant

The Vandermonde determinant is the determinant of the matrix whose rows are successive powers of a list of numbers x_1, ..., x_n. Remarkably, it factors into the product of all pairwise differences (x_j - x_i) over i < j. That closed form is one of the most quoted determinant identities, and it is the algebraic engine behind polynomial interpolation.

Concretely, the Vandermonde matrix V has entry V_{ij} = x_i^(j-1), so its columns run 1, x, x^2, ..., x^(n-1) evaluated at each node. Then det(V) = product over all pairs i < j of (x_j - x_i). The clean way to see it: det(V) is a polynomial in the x's that vanishes whenever two nodes coincide (two equal rows make det = 0), so each factor (x_j - x_i) must divide it; matching degree and leading coefficient pins the product exactly.

Why it matters: det(V) is nonzero if and only if the nodes x_1, ..., x_n are all distinct. That single statement says the interpolation problem 'find the unique polynomial of degree < n through n given points' always has a unique solution when the x-coordinates differ — because solving for the coefficients is solving the Vandermonde linear system, which is invertible exactly when its determinant is nonzero. The same fact certifies that 1, x, ..., x^(n-1) are linearly independent functions.

Caveat: the Vandermonde matrix is a beloved theoretical object but notoriously ill-conditioned for large n — its determinant can be enormous or minuscule, and numerically inverting it to do interpolation is unstable. In practice one uses Lagrange or Newton interpolation formulas, which sidestep forming and inverting V while exploiting the same distinctness guarantee.