Subsection
{\sc\large This Section is a Draft, Subject to Changes}
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Alexandre-Th\'{e}ophile Vandermonde was a French mathematician in the 1700's who was among the first to write about basic properties of the determinant (such as the effect of swapping two rows). However, the determinant that bears his name (Theorem
DVM) does not appear in any of his four published mathematical papers.
Definition VM Vandermonde Matrix
An square matrix of size $n$, $A$, is a Vandermonde matrix if there are scalars, $\scalarlist{x}{n}$ such that $\matrixentry{A}{ij}=x_{i}^{j-1}$, $1\leq i\leq n$, $1\leq j\leq n$.
$\square$
Example VM4 Vandermonde matrix of size 4
Vandermonde matrices are not very interesting as numerical matrices, but instead appear more often in proofs and applications where the scalars $x_i$ are carried as symbols. Two such applications are in the sections on secret-sharing (Section
SAS) and curve-fitting (Section
CF). Principally, we would like to know when Vandermonde matrices are nonsingular, and the most convenient way to check this is by determining when the determinant is nonzero (Theorem
SMZD). As a bonus, the determinant of a Vandermonde matrix has an especially pleasing formula.
Theorem DVM Determinant of a Vandermonde Matrix
Suppose that $A$ is a Vandermonde matrix of size $n$ built with the scalars $\scalarlist{x}{n}$. Then
\begin{align*}
\detname{A}
&=\prod_{1\leq i<j\leq n}\left(x_j-x_i\right)
\end{align*}
Proof
Before we had Theorem
DVM we could see that if two of the scalar values were equal, then the Vandermonde matrix would have two equal rows and hence be singular (Theorem
DERC, Theorem
SMZD). But with this expression for the determinant, we can establish the converse.
Theorem NVM Nonsingular Vandermonde Matrix
A Vandermonde matrix of size $n$ with scalars $\scalarlist{x}{n}$ is nonsingular if and only if the scalars are all different.
Proof
