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, $q\times q$ say, in which case the $i$-th column is given by
$u(z_i)$, where we write $u(z)=(1,z,...,z^{q-1})^\top$. If all the $z_i$
($i=1,...,q$) are different, the Vandermonde matrix is non-singular, otherwise
not. The latter case obviously takes place when all $z_i$ 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 $(i-1)$-th derivative $u^{(i-1)}(z)^\top$.
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 $w(z)=(1,z,...,z^{r-1})$, 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