Vandermonde-Matrix

Under a Vandermonde matrix (after A.-T. Vandermonde ) is understood in mathematics, a matrix that has a special form described below.

For a tuple of real numbers, or more generally of elements in a body is the Vandermonde matrix defined by:

The determinant is also called Vandermonde determinant, it has the value

In particular, the Vandermonde matrix is regular if the are pairwise different.

Application: polynomial

The Vandermonde matrix plays in the interpolation of functions play an important role: If the function values ​​are to be interpolated by a polynomial of degree at the support points, then performs the approach

To the linear system of equations

With a Vandermonde matrix as coefficient matrix. Out of the above property of the Vandermonde determinant, therefore, follows in particular that the interpolation problem has a unique solution if and only if all nodes are different in pairs.

In the standard basis of the polynomials, the matrix is ​​, however, very ill-conditioned and the resolution of standard methods with a term in quite expensive, which is why one chooses different representations for the polynomials. For details on polynomial interpolation and down.

Other properties

The Vandermonde matrix of the above system of equations diagonalizing the companion matrix of the polynomial, we have:

For large numbers the system of equations above can be solved via the following relationship by which the inverse of the Vandermonde matrix is closely connected with its transposed. Through the implementation of polynomial coefficients forming the Hankel matrix

And the diagonal matrix. If all nodes are different in pairs, is regular. This is true