Vandermonde Matrix

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 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 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.

When two or more α_i are equal, the corresponding polynomial interpolation problem (see below) is underdetermined. In that case one may use a generalization called confluent Vandermonde matrices, which makes the matrix non-singular while retaining most properties. If α_i = α_{i + 1} = ... = α_i+k and α_i ≠ α_{i − 1}, then the (i + k)th row is given by

The above formula for confluent Vandermonde matrices can be readily derived by letting two parameters and go arbitrarily close to each other. The difference vector between the rows corresponding to and scaled to a constant yields the above equation (for k = 1). Similarly, the cases k > 1 are obtained by higher order differences. Consequently, the confluent rows are derivatives of the original Vandermonde row.