Let $M(n)$ be the algebra (viewed as both a Lie and an associative algebra) of $n\times n$ matrices over $\mathbb{C}$. Let $P(n)$ denote the algebra of polynomials on $M(n)$. The associative commutator on the universal enveloping algebra induces a Poisson structure on $P(n)$. Let $J(n)$ be the commutative Poisson subalgebra of $P(n)$ generated by the invariants $P(m)^{Gl(m)} \text{ for } m=1,\cdots , n$. $J(n)$ gives rise to a commutative Lie algebra of vector fields on $M(n)$; $V=\{ \xi_{f}| f\in J(n)\}$. These fields integrate to an action of a commutative, simply connected complex analytic group $A\simeq\mathbb{C} ^{\frac{(n-1)n}{2}}$ on $M(n)$. Note that the dimension of this group is exactly half the dimension of the generic coadjoint orbits. Moreover, on the most generic orbits of $A$, the commutative Lie algebra $V$ is an algebra of symplectic vector fields of exactly half the dimension of the generic coadjoint orbits. We will discuss the orbit structure of the action of $A$ on $M(n)$. We will give a description of the work of Kostant-Wallach in the most generic case in such a form that can be used to establish a formalism for dealing with less generic orbits.