Definition of Vandermonde matrix

Definition: A Vandermonde matrix is a matrix where each row is a geometric progression $1, \alpha_i, \alpha_i^2, \ldots, \alpha_i^{n-1}$ for some numbers $\alpha_i$:

$$V = \begin{pmatrix} 1 & \alpha_1 & \alpha_1^2 & \cdots & \alpha_1^{n-1} \\ 1 & \alpha_2 & \alpha_2^2 & \cdots & \alpha_2^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \alpha_m & \alpha_m^2 & \cdots & \alpha_m^{n-1} \end{pmatrix}$$

Key property: The determinant of a square Vandermonde matrix is:

$$\det(V) = \prod_{1 \leq i < j \leq n} (\alpha_j - \alpha_i)$$

$V$ is invertible iff all $\alpha_i$ are distinct.

Applications:

• Polynomial interpolation: solving $Vc = y$ finds coefficients of a polynomial passing through given points
• Coding theory and error-correcting codes
• Signal processing

Vandermonde matrices can be ill-conditioned when the $\alpha_i$ values are close together.