Generalized Vandermonde Matrix

A generalized Vandermonde matrix of two sequences a and b where a is an increasing sequence of positive integers and b is an increasing sequence of nonnegative integers of the same length is the outer product of a and b with multiplication operation given by the power function. The generalized Vandermonde matrix can be implemented in the Wolfram Language as

  Vandermonde[a_List?VectorQ, b_List?VectorQ] :=
    Outer[Power, a, b] /; Equal @@ Length /@ {a, b}

A generalized Vandermonde matrix is a minor of a Vandermonde matrix. Alternatively, it has the same form as a Vandermonde matrix (a_(ij))=a_i^(b_j), where a_i is an increasing sequence of positive integers, except now b_j is any increasing sequence of nonnegative integers. In the special case of a Vandermonde matrix, b_j=j-1.

While there is no general formula for the determinant of a generalized Vandermonde matrix, its determinant is always positive. Since any minor of a generalized Vandermonde matrix is also a generalized Vandermonde matrix, they are always totally positive.

See also

Vandermonde Matrix

This entry contributed by Todd Rowland

Explore with Wolfram|Alpha


Gantmacher, F. R. The Theory of Matrices, Vol. 2. Providence, RI: AMS Chelsea, 2000.

Referenced on Wolfram|Alpha

Generalized Vandermonde Matrix

Cite this as:

Rowland, Todd. "Generalized Vandermonde Matrix." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications