matrixcalc (version 1.0-6)

vandermonde.matrix: Vandermonde matrix

Description

This function returns an m by n matrix of the powers of the alpha vector

Usage

vandermonde.matrix(alpha, n)

Value

A matrix.

Arguments

alpha

A numerical vector of values

n

The column dimension of the Vandermonde matrix

Author

Frederick Novomestky

Details

In linear algebra, a Vandermonde matrix is an \(m \times n\) matrix with terms of a geometric progression of an \(m \times 1\) parameter vector \({\bf{\alpha }} = {\left\lbrack {\begin{array}{cccc} {{\alpha _1}}&{{\alpha _2}}& \cdots &{{\alpha _m}} \end{array}} \right\rbrack^\prime }\)

such that \(V\left( {\bf{\alpha }} \right) = \left\lbrack {\begin{array}{ccccc} 1&{{\alpha _1}}&{\alpha _1^2}& \cdots &{\alpha _1^{n - 1}}\\ 1&{{\alpha _2}}&{\alpha _2^2}& \cdots &{\alpha _2^{n - 1}}\\ 1&{{\alpha _3}}&{\alpha _3^2}& \cdots &{\alpha _3^{n - 1}}\\ \cdots & \cdots & \cdots & \cdots & \cdots \\ 1&{{\alpha _m}}&{\alpha _m^2}& \cdots &{\alpha _m^{n - 1}} \end{array}} \right\rbrack\).

References

Horn, R. A. and C. R. Johnson (1991). Topics in matrix analysis, Cambridge University Press.

Examples

alpha <- c( .1, .2, .3, .4 )
V <- vandermonde.matrix( alpha, 4 )
print( V )