 TOPICS  # Generalized Vandermonde Matrix

A generalized Vandermonde matrix of two sequences and where is an increasing sequence of positive integers and is an increasing sequence of nonnegative integers of the same length is the outer product of and 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 , where is an increasing sequence of positive integers, except now is any increasing sequence of nonnegative integers. In the special case of a Vandermonde matrix, .

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.

Vandermonde Matrix

This entry contributed by Todd Rowland

## Explore with Wolfram|Alpha ## References

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. https://mathworld.wolfram.com/GeneralizedVandermondeMatrix.html