Vandermonde matrices are well known. They have a number of interesting
properties and play a role in (Lagrange) interpolation problems, partial
fraction expansions, and finding solutions to linear ordinary differential
equations, to mention just a few applications. Usually, one takes these
matrices square, qΓq say, in which case the i-th column is given by
u(ziβ), where we write u(z)=(1,z,...,zqβ1)β€. If all the ziβ
(i=1,...,q) are different, the Vandermonde matrix is non-singular, otherwise
not. The latter case obviously takes place when all ziβ are the same, z
say, in which case one could speak of a confluent Vandermonde matrix.
Non-singularity is obtained if one considers the matrix V(z) whose i-th
column (i=1,...,q) is given by the (iβ1)-th derivative u(iβ1)(z)β€.
We will consider generalizations of the confluent Vandermonde matrix V(z)
by considering matrices obtained by using as building blocks the matrices
M(z)=u(z)w(z), with u(z) as above and w(z)=(1,z,...,zrβ1), together
with its derivatives M(k)(z). Specifically, we will look at matrices whose
ij-th block is given by M(i+j)(z), where the indices i,j by convention
have initial value zero. These in general non-square matrices exhibit a
block-Hankel structure. We will answer a number of elementary questions for
this matrix. What is the rank? What is the null-space? Can the latter be
parametrized in a simple way? Does it depend on z? What are left or right
inverses? It turns out that answers can be obtained by factorizing the matrix
into a product of other matrix polynomials having a simple structure. The
answers depend on the size of the matrix M(z) and the number of derivatives
M(k)(z) that is involved. The results are obtained by mostly elementary
methods, no specific knowledge of the theory of matrix polynomials is needed