Editing Vandermonde matrix (section)

{{Short description|Mathematical concept}}
In [[linear algebra]], a '''Vandermonde matrix''', named after [[Alexandre-Théophile Vandermonde]], is a [[matrix (mathematics)|matrix]] with the terms of a [[geometric progression]] in each row: an <math>(m + 1) \times (n + 1)</math> matrix
:<math>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}</math>
with entries <math>V_{i,j} = x_i^j </math>, the ''j''<sup>th</sup> power of the number <math>x_i</math>, for all [[Zero-based numbering|zero-based]] indices <math>i </math> and <math>j </math>.<ref>Roger A. Horn and Charles R. Johnson (1991), ''Topics in matrix analysis'', Cambridge University Press. ''See Section 6.1''.</ref> Some authors define the Vandermonde matrix as the [[transpose]] of the above matrix.<ref name=":0">{{Cite book |last1=Golub |first1=Gene H. |title=Matrix Computations |last2=Van Loan |first2=Charles F. |publisher=The Johns Hopkins University Press |year=2013 |isbn=978-1-4214-0859-0 |edition=4th |pages=203–207}}</ref><ref name="MS" />

The [[determinant]] of a [[square matrix|square]] Vandermonde matrix (when <math>n=m</math>) is called a '''Vandermonde determinant''' or [[Vandermonde polynomial]]. Its value is:
:<math>\det(V) = \prod_{0 \le i < j \le m} (x_j - x_i). </math>
This is non-zero if and only if all <math>x_i</math> are distinct (no two are equal), making the Vandermonde matrix [[Invertible matrix|invertible]].