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 is of the form
(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
Vandermonde matrix equation requires 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. Referenced
on Wolfram|Alpha Vandermonde Matrix
Cite this as:
Weisstein, Eric W. "Vandermonde Matrix."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/VandermondeMatrix.html
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