Vandermonde matrix

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

or

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

Vandermonde matrices are named after Alexandre-Théophile Vandermonde.

These matrices are useful in polynomial interpolation, since solving the system of linear equations Vu=y for u is equivalent to finding the coefficients u_j of a polynomial that has values y_i at α_i.

The determinant of an Vandermonde matrix can be expressed as follows:

If m≤n, then the matrix V has maximum rank (m) if and only if all α_i are distinct.

When two or more α_i are equal, the corresponding polynomial interpolation problem is ill-posed. In that case one may use a generalisation called confluent Vandermonde matrices. If α_i = α_i+1 = ... = α_i+k and α_i ≠ α_i-1, then the (i + k)th row is given by

Confluent Vandermonde matrices are used in Hermite interpolation.

See also:

• Lagrange polynomial

Reference

Roger A. Horn and Charles R. Johnson (1991). Topics in matrix analysis, Section 6.1. Cambridge University Press.