  We develop the algebraic instance of an algorithmic approach to the matricial
Hausdorff moment problem on a compact interval $[\alpha,\beta]$ of the real
axis. Our considerations are along the lines of the classical Schur algorithm
and the treatment of the Hamburger moment problem on the real axis by
Nevanlinna. More precisely, a transformation of matrix sequences is
constructed, which transforms Hausdorff moment sequences into Hausdorff moment
sequences reduced by 1 in length. It is shown that this transformation
corresponds essentially to the left shift of the associated sequences of
canonical moments. As an application, we show that a matricial version of the
arcsine distribution can be used to characterize a certain centrality property
of non-negative Hermitian measures on $[\alpha,\beta]$.
