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Vandermonde Matrix — Definition, Formula & Examples

A Vandermonde matrix is a matrix where each row (or column) consists of successive powers of a single value, so the entry in row ii, column jj equals xi jx_i^{\,j}. It appears naturally in polynomial interpolation and has a determinant given by a compact product formula.

Given distinct scalars x0,x1,…,xn−1x_0, x_1, \ldots, x_{n-1}, the n×nn \times n Vandermonde matrix VV is defined by Vij=xi jV_{ij} = x_i^{\,j} for 0≤i,j≤n−10 \le i,j \le n-1. Its determinant equals det⁡(V)=∏0≤i<j≤n−1(xj−xi)\det(V) = \prod_{0 \le i < j \le n-1}(x_j - x_i), which is nonzero if and only if all xix_i are distinct.

Key Formula

V=(1x0x02⋯x0n−11x1x12⋯x1n−1⋮⋮⋮⋱⋮1xn−1xn−12⋯xn−1n−1)V = \begin{pmatrix} 1 & x_0 & x_0^2 & \cdots & x_0^{n-1} \\ 1 & x_1 & x_1^2 & \cdots & x_1^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_{n-1} & x_{n-1}^2 & \cdots & x_{n-1}^{n-1} \end{pmatrix}
Where:
  • xix_i = The distinct scalars (nodes) that generate the matrix
  • nn = The number of nodes, giving an n × n matrix
  • VijV_{ij} = Entry in row i, column j, equal to x_i raised to the j-th power

How It Works

To build a Vandermonde matrix, list your values x0,x1,…,xn−1x_0, x_1, \ldots, x_{n-1} and form each row by raising xix_i to the powers 0,1,2,…,n−10, 1, 2, \ldots, n-1. The resulting system Vc=yV\mathbf{c} = \mathbf{y} finds the coefficients of the unique polynomial of degree at most n−1n-1 passing through given data points. Because the determinant is a product of all pairwise differences, the matrix is invertible precisely when no two xix_i values coincide.

Worked Example

Problem: Construct the 3 × 3 Vandermonde matrix for the nodes x₀ = 1, x₁ = 2, x₂ = 3, and compute its determinant.
Build the matrix: Each row uses powers 0, 1, 2 of the corresponding node.
V=(111124139)V = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \end{pmatrix}
Apply the determinant formula: The Vandermonde determinant is the product of all differences (x_j − x_i) for i < j.
det⁡(V)=(x1−x0)(x2−x0)(x2−x1)=(2−1)(3−1)(3−2)=1⋅2⋅1=2\det(V) = (x_1 - x_0)(x_2 - x_0)(x_2 - x_1) = (2-1)(3-1)(3-2) = 1 \cdot 2 \cdot 1 = 2
Answer: The Vandermonde matrix is (111124139)\begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \end{pmatrix} with determinant 22.

Why It Matters

Vandermonde matrices are central to polynomial interpolation (Lagrange and Newton methods) and appear in coding theory, signal processing (DFT matrices are complex Vandermonde matrices), and curve fitting. Understanding when the determinant is nonzero tells you exactly when a unique interpolating polynomial exists.

Common Mistakes

Mistake: Using rows of powers starting at 1 instead of 0, producing columns of xi1,xi2,…x_i^1, x_i^2, \ldots and omitting the column of ones.
Correction: The first column must be xi0=1x_i^0 = 1 for every row. Without it, the matrix does not represent the standard Vandermonde form and the determinant formula does not apply.