# Definition of Vandermonde matrix

Put content here**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 $\alpha_i$:
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$$V = \begin{pmatrix} 1 & \alpha_1 & \alpha_1^2 & \cdots & \alpha_1^{n-1} \\ 1 & \alpha_2 & \alpha_2^2 & \cdots & \alpha_2^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \alpha_m & \alpha_m^2 & \cdots & \alpha_m^{n-1} \end{pmatrix}$$
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**Key property:** The determinant of a square Vandermonde matrix is:
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$$\det(V) = \prod_{1 \leq i < j \leq n} (\alpha_j - \alpha_i)$$
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$V$ is invertible iff all $\alpha_i$ are distinct.
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**Applications:**
- Polynomial interpolation: solving $Vc = y$ finds coefficients of a polynomial passing through given points
- Coding theory and error-correcting codes
- Signal processing
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Vandermonde matrices can be ill-conditioned when the $\alpha_i$ values are close together.

# Parents

* Particular types of matrices