Vandermonde matrix

Remarks

Example: for $x_1 = 1$, $x_2 = 2$, $x_3 = 3$ and $n = 3$, the matrix is $\begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \end{pmatrix}$; row $i$ is $(1, x_i, x_i^2)$. Non-example: $\begin{pmatrix} 2 & 4 \\ 3 & 9 \end{pmatrix}$ is not a Vandermonde matrix, because a Vandermonde matrix has first column $1, \dots, 1$. It is obtained from the Vandermonde matrix $\begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix}$ of $2, 3$ by multiplying row $i$ by $x_i$. In the square case $m = n$, the Vandermonde determinant is $\det V = \prod_{1 \le i < j \le n} (x_j - x_i)$. For a polynomial $p(t) = c_0 + c_1 t + \cdots + c_{n-1} t^{n-1}$, the vector $V c$ lists the values $p(x_1), \dots, p(x_m)$. Some authors use the transpose of $V$ instead.