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 submatrix 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, 
.
For 
rows and columns, the determinant has the formula
 |
where 
is the Schur polynomial for the partition 
,
.
The Schur polynomial is positive
for positive 
, so the determinant is always
positive. Since any square submatrix
of a generalized Vandermonde matrix is also a generalized Vandermonde matrix,
they are always totally positive.
