  There is a standard way to define two symplectic (hamiltonian) structures,
the first and second Gelfand-Dikii brackets, on the space of ordinary linear
differential operators of order $m$, $L = -d^m + U_1 d^{m-1} + U_2 d^{m-2} +
\ldots + U_m$. In this paper, I consider in detail the case where the $U_k$ are
$n\times n$-matrix-valued functions, with particular emphasis on the (more
interesting) second Gelfand-Dikii bracket. Of particular interest is the
reduction to the symplectic submanifold $U_1=0$. This reduction gives rise to
matrix generalizations of (the classical version of) the {\it non-linear}
$W_m$-algebras, called $V_{m,n}$-algebras. The non-commutativity of the
matrices leads to {\it non-local} terms in these $V_{m,n}$-algebras. I show
that these algebras contain a conformal Virasoro subalgebra and that
combinations $W_k$ of the $U_k$ can be formed that are $n\times n$-matrices of
conformally primary fields of spin $k$, in analogy with the scalar case $n=1$.
In general however, the $V_{m,n}$-algebras have a much richer structure than
the $W_m$-algebras as can be seen on the examples of the {\it non-linear} and
{\it non-local} Poisson brackets of any two matrix elements of $U_2$ or $W_3$
which I work out explicitly for all $m$ and $n$. A matrix Miura transformation
is derived, mapping these complicated second Gelfand-Dikii brackets of the
$U_k$ to a set of much simpler Poisson brackets, providing the analogue of the
free-field realization of the $W_m$-algebras.
