Vandermonde determinant

In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row, i.e., an m × n matrix

or

for all indices i and j. (Some authors use the transpose of the above matrix.)

The determinant of a square Vandermonde matrix (where m = n) can be expressed as

This is called the Vandermonde determinant or Vandermonde polynomial. If all the numbers ${\displaystyle \alpha _{i}}$ are distinct, then it is non-zero.

The Vandermonde determinant is sometimes called the discriminant, although many sources, including this article, refer to the discriminant as the square of this determinant. Note that the Vandermonde determinant is alternating in the entries, meaning that permuting the ${\displaystyle \alpha _{i}}$ by an odd permutation changes the sign, while permuting them by an even permutation does not change the value of the determinant. It thus depends on the order, while its square (the discriminant) does not depend on the order.

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