Vandermonde matrix

In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an ${\displaystyle (m+1)\times (n+1)}$ matrix

${\displaystyle V=V(x_{0},x_{1},\cdots ,x_{m})={\begin{bmatrix}1&x_{0}&x_{0}^{2}&\dots &x_{0}^{n}\\1&x_{1}&x_{1}^{2}&\dots &x_{1}^{n}\\1&x_{2}&x_{2}^{2}&\dots &x_{2}^{n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&x_{m}&x_{m}^{2}&\dots &x_{m}^{n}\end{bmatrix}}}$

with entries ${\displaystyle V_{i,j}=x_{i}^{j}}$, the j^th power of the number ${\displaystyle x_{i}}$, for all zero-based indices ${\displaystyle i}$ and ${\displaystyle j}$.^[1] Some authors define the Vandermonde matrix as the transpose of the above matrix.^[2][3]

The determinant of a square Vandermonde matrix (when ${\displaystyle n=m}$) is called a Vandermonde determinant or Vandermonde polynomial. Its value is:

${\displaystyle \det(V)=\prod _{0\leq i<j\leq m}(x_{j}-x_{i}).}$

This is non-zero if and only if all ${\displaystyle x_{i}}$ are distinct (no two are equal), making the Vandermonde matrix invertible.