Vandermonde matrix

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

$V={\begin{bmatrix}1&\alpha _{1}&\alpha _{1}^{2}&\dots &\alpha _{1}^{n-1}\\1&\alpha _{2}&\alpha _{2}^{2}&\dots &\alpha _{2}^{n-1}\\1&\alpha _{3}&\alpha _{3}^{2}&\dots &\alpha _{3}^{n-1}\\\vdots &\vdots &\vdots &&\vdots \\1&\alpha _{m}&\alpha _{m}^{2}&\dots &\alpha _{m}^{n-1}\\\end{bmatrix}}$

or

$V_{i,j}=\alpha _{i}^{j-1}$

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 $n\times n$ Vandermonde matrix can be expressed as:

$\det(V)=\prod _{1\leq i<j\leq n}(\alpha _{j}-\alpha _{i})=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )\alpha _{1}^{\sigma (1)-1}...\alpha _{n}^{\sigma (n)-1}$

This determinant is sometimes called the discriminant, although many authors, including Wikipedia, refer to the discriminant as the square of the determinant.

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, which makes the matrix positive definite 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

$V_{i+k,j}={\begin{cases}0,&{\mbox{if }}j\leq k;\\{\frac {(j-1)!}{(j-k-1)!}}\alpha _{i}^{j-k-1},&{\mbox{if }}j>k.\end{cases}}$

The above formula for confluent Vandermonde matrices can be readily derived by letting two parameters $\alpha _{i}$ and $\alpha _{j}$ go arbitrarly close to each other. The difference vector between the rows corresponding to $\alpha _{i}$ and $\alpha _{j}$ 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.

Confluent Vandermonde matrices are used in Hermite interpolation.

See also

• Lagrange polynomial
• Wronskian

Reference

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