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Vandermonde Matrix

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A Vandermonde matrix is a type of matrix that arises in the polynomial least squares fitting, Lagrange interpolating polynomials (Hoffman and Kunze p. 114), and the reconstruction of a statistical distribution from the distribution's moments (von Mises 1964; Press et al. 1992, p. 83). A Vandermonde matrix of order n is of the form

[1 x_1 x_1^2 ... x_1^(n-1); 1 x_2 x_2^2 ... x_2^(n-1); | | | ... |; 1 x_n x_n^2 ... x_n^(n-1)].

(Press et al. 1992; Meyer 2000, p. 185). A Vandermonde matrix is sometimes also called an alternant matrix (Marcus and Minc 1992, p. 15). Note that some authors define the transpose of this matrix as the Vandermonde matrix (Marcus and Minc 1992, p. 15; Golub and Van Loan 1996; Aldrovandi 2001, p. 193).

The solution of an n×n Vandermonde matrix equation requires O(n^2) operations. The determinants of Vandermonde matrices have a particularly simple form.

See also

Generalized Vandermonde Matrix, Least Squares Fitting--Polynomial, Toeplitz Matrix, Tridiagonal Matrix, Vandermonde Determinant

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References

Aldrovandi, R. Special Matrices of Mathematical Physics: Stochastic, Circulant and Bell Matrices. Singapore: World Scientific, 2001.Golub, G. H. and Van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, 1996.Hoffman, K. M. and Kunze, R. Linear Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice Hall, 1971.Marcus, M. and Minc, H. "Vandermonde Matrix." §2.6.2 in A Survey of Matrix Theory and Matrix Inequalities. New York: Dover, pp. 15-16, 1992.Meyer, C. D. Matrix Analysis and Applied Linear Algebra. Philadelphia, PA: SIAM, 2000.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Vandermonde Matrices and Toeplitz Matrices." §2.8 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 82-89, 1992.Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, pp. 56-57, 2004. http://www.mathematicaguidebooks.org/.von Mises, R. Mathematical Theory of Probability and Statistics. New York: Academic Press, 1964.

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Vandermonde Matrix

Cite this as:

Weisstein, Eric W. "Vandermonde Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/VandermondeMatrix.html

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