nLab Vandermonde determinant

Definition

The Vandermonde determinant or Vandermonde polynomial $\Delta(x_1,\ldots,x_n)$ is the determinant of the Vandermonde matrix:

(1)$$V(x_1,\ldots,x_n) \;\coloneqq\; \left( \array{ 1 & x_1 & \cdots & x_1^{n-1} \\ 1 & x_2 &\cdots & x_2^{n-1} \\ \cdot &\cdot &\cdot &\cdots \\ 1 & x_n &\cdots &x_n^{n-1} } \right) \mathrlap{\,.}$$

Proposition

$$\Delta(x_1,\ldots,x_n) \;=\; \prod_{1\leq i\lt j\leq n} (x_j - x_i) \,.$$

Proof

We argue by induction. Without loss of generality, assume $x_1, \ldots, x_{n-1}$ are distinct (else the value is zero as claimed), and treat these as constants. Write

$$\det \left( \array{1 & x_1 & \cdots & x_1^{n-1}\\ 1 & x_2 &\cdots & x_2^{n-1}\\ \cdot &\cdot &\cdot &\cdots\\ 1 & x &\cdots &x^{n-1}}\right) = f(x) = \sum_{i=1}^{n-1} a_i x^i.$$

Then the leading coefficient $a_{n-1}$ is $\prod_{1\leq i\lt j\leq n-1} (x_j - x_i)$ by inductive hypothesis, and $f(x)$ has simple roots at $x = x_1, \ldots, x_{n-1}$ since each of those values makes two rows equal (hence with zero determinant). It follows that

$$\begin{array}{ccl} f(x) &=& a_{n-1} \, \textstyle{\prod_{i=1}^{n-1}} (x - x_i) \\ &=& \left( \textstyle{\prod_{1\leq i\lt j\leq n-1}} (x_j - x_i) \right) \textstyle{\prod_{i=1}^{n-1}} (x - x_i) \,. \end{array}$$

and the inductive step follows by setting $x = x_n$.

Properties

Remark

If follows for instance that the product $\prod_{1\leq i\lt j\leq n} (x_j - x_i)$ changes sign when any pair $(x_r, x_s)$ of the variables are interchanged, because this corresponds to swapping the corresponding rows of the Vandermonde matrix (1), and because the determinant is alternating multilinear.

This property is important, for instance, when understanding the Vandermonde determinant as a factor in the Laughlin wavefunction ground states of fractional quantum Hall systems at odd filling-fraction, where it reflects the skew-symmetry required of many-fermion (spin-polarized electron) wavefunctions.

Applications

The Vandermonde determinant appears in many important situations, as a square root of a discriminant, sometimes as a Wronskian, and also in Lagrange interpolation (see wikipedia Lagrange polynomial), Selberg integral etc.

Literature

• Wikipedia: Vandermonde matrix

• Alexander Varchenko: Multidimensional Vandermonde Determinant, Uspekhi Mat. Nauk 43 4 (1988) 190.

• Alexander Varchenko, Beta-function of Euler, Vandermonde determinant, Legendre equation and critical values of linear functions of configuration of hyperplanes, I. Izv. Akademii Nauk USSR, Seriya Mat. 53 6 (1989) 1206-1235.