Vandermonde matrix

A matrix built entirely from powers of one set of values -- 1, x, x^2, and so on -- whether solved exactly (Chapter 3) or fit by least squares (Chapter 9).

A Vandermonde matrix is a matrix whose columns (or rows, depending on convention) are successive powers of a fixed set of values $x_0,\ldots,x_n$ . Chapter 3 met it as the coefficient matrix behind the interpolating polynomial's existence-uniqueness proof; Chapter 9 reuses the exact same object as the design matrix $A$ in $A\mathbf{x}\approx\mathbf{b}$ for fitting a degree- $d$ polynomial $y=c_0+c_1x+\cdots+c_dx^d$ to $n>d+1$ data points by least squares.

Examples

For nodes $1,2,3$ and degree 2, the Vandermonde matrix is $\begin{pmatrix}1&1&1\\1&2&4\\1&3&9\end{pmatrix}$ -- each row is $1,x_i,x_i^2$ .