|
In linear algebra, a Vandermonde matrix is a matrix with a geometric progression in each row, i.e;
- <math>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}<math>
or
- <math>V_{i,j} = \alpha_i^{j-1}<math>
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 uj of a polynomial that has values yi at αi.
The determinant of an <math>n\times n<math> Vandermonde matrix can be expressed as follows:
- <math>\det(V) = \prod_{1\le i
If m≤n, then the matrix V has maximum rank (m) if and only if all αi are distinct.
When two or more αi 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 αi = αi+1 = ... = αi+k and αi ≠ αi-1, then the (i + k)th row is given by
- <math> V_{i+k,j} = \begin{cases} 0, & \mbox{if } j \le k; \\ \frac{(j-1)!}{(j-k-1)!} \alpha_i^{j-k-1}, & \mbox{if } j > k. \end{cases} <math>
The above formula for confluent Vandermonde matrices can be readily derived by letting two parameters <math>\alpha_i<math> and <math>\alpha_j<math> go arbitrarly close to each other. The difference vector between the rows corresponding to <math>\alpha_i<math> and <math>\alpha_j<math> 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
Reference
Roger A. Horn and Charles R. Johnson (1991). Topics in matrix analysis, Section 6.1. Cambridge University Press.
| 
