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The Vandermonde determinant or Vandermonde polynomial is the determinant of the Vandermonde matrix:
We argue by induction. Without loss of generality, assume are distinct (else the value is zero as claimed), and treat these as constants. Write
Then the leading coefficient is by inductive hypothesis, and has simple roots at since each of those values makes two rows equal (hence with zero determinant). It follows that
and the inductive step follows by setting .
If follows for instance that the product changes sign when any pair 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.
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.
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.
Last revised on January 8, 2025 at 14:10:48. See the history of this page for a list of all contributions to it.