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 ( m + 1 ) × ( n + 1 ) {\displaystyle (m+1)\times (n+1)} matrix V = V ( x 0 , x 1 , ⋯ , x m ) = ( 1 x 0 x 0 2 … x 0 n 1 x 1 x 1 2 … x 1 n 1 x 2 x 2 2 … x 2 n ⋮ ⋮ ⋮ ⋱ ⋮ 1 x m x m 2 … x m n ) {\displaystyle V=V(x_{0},x_{1},\cdots ,x_{m})={\begin{pmatrix}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{pmatrix}}} with entries V i , j = x i j {\displaystyle V_{i,j}=x_{i}^{j}} , the jth power of the number x i {\displaystyle x_{i}} , for all zero-based indices i {\displaystyle i} and j {\displaystyle j} . Some authors define the Vandermonde matrix as the transpose of the above matrix. The determinant of a square Vandermonde matrix (when n = m {\displaystyle n=m} ) is called a Vandermonde determinant or Vandermonde polynomial. Its value is: det ( V ) = ∏ 0 ≤ i < j ≤ n ( x j − x i ) = ( − 1 ) n ( n + 1 ) / 2 ∏ 0 ≤ i < j ≤ n ( x i − x j ) . {\displaystyle \det(V)=\prod _{0\leq i<j\leq n}(x_{j}-x_{i})=(-1)^{n(n+1)/2}\prod _{0\leq i<j\leq n}(x_{i}-x_{j}).} This is non-zero if and only if all x i {\displaystyle x_{i}} are distinct (no two are equal), making the Vandermonde matrix invertible.