  In this paper we obtain a description of the Hermitian operators acting on
the Hilbert space $\C^n$, description which gives a complete solution to the
over parameterization problem. More precisely we provide an explicit
parameterization of arbitrary $n$-dimensional operators, operators that may be
considered either as Hamiltonians, or density matrices for finite-level quantum
systems. It is shown that the spectral multiplicities are encoded in a flag
unitary matrix obtained as an ordered product of special unitary matrices, each
one generated by a complex $n-k$-dimensional unit vector, $k=0,1,...,n-2$. As a
byproduct, an alternative and simple parameterization of Stiefel and Grassmann
manifolds is obtained.
