Vandermonde matrix

In linear algebra, a Vandermonde matrix is a matrix with a geometric progression in each column, i.e;

In mathematical terms:

These matrices are useful in polynomial interpolation, since solving an equation for , is equivalent to finding the coefficents of a polynomial that has values at .

The determinant of a square Vandermonde matrix of a dimension can be expressed as follows:

If two or more exponents are equal, the rank of the matrix decreases (if all are distinct, then is of full rank). This problem can alleviated by using a generalisation called confluent Vandermonde matrices, where the k-multiple columns are replaced by:

where

Vandermonde matrices were named after Alexandre-Théophile Vandermonde (1735-1796), a French mathematician and musician.