Vandermonde Matrix - Applications

Applications

The Vandermonde matrix evaluates a polynomial at a set of points; formally, it transforms coefficients of a polynomial to the values the polynomial takes at the points The non-vanishing of the Vandermonde determinant for distinct points shows that, for distinct points, the map from coefficients to values at those points is a one-to-one correspondence, and thus that the polynomial interpolation problem is solvable with unique solution; this result is called the unisolvence theorem.

They are thus useful in polynomial interpolation, since solving the system of linear equations Vu = y for u with V an m × n Vandermonde matrix is equivalent to finding the coefficients u_j of the polynomial(s)

of degree ≤ n − 1 which has (have) the property

The Vandermonde matrix can easily be inverted in terms of Lagrange basis polynomials: each column is the coefficients of the Lagrange basis polynomial, with terms in increasing order going down. The resulting solution to the interpolation problem is called the Lagrange polynomial.

The Vandermonde determinant plays a central role in the Frobenius formula, which gives the character of conjugacy classes of representations of the symmetric group.

When the values range over powers of a finite field, then the determinant has a number of interesting properties: for example, in proving the properties of a BCH code.

Confluent Vandermonde matrices are used in Hermite interpolation.

A commonly known special Vandermonde matrix is the discrete Fourier transform matrix (DFT matrix), where the numbers α_i are chosen to be the m different mth roots of unity.

The Vandermonde matrix diagonalizes a companion matrix.