  This article presents an investigation on the global hypoellipticity problem
for systems belonging to the class $P = D_t + Q(t,D_x)$, where $Q(t,D_x)$ is a
$m\times m$ matrix with entries $c_{j,k}(t)Q_{j,k}(D_x)$. The coefficients
$c_{j,k}(t)$ are smooth, complex-valued functions on the torus $\mathbb{T}
\simeq \mathbb{R}/2\pi\mathbb{Z}$ and $Q_{j,k}(D_x)$ are pseudo-differential
operators on $ \mathbb{T}^n$. The approach consists in establishing conditions
on the matrix symbol $Q(t,\xi)$ such that it can be transformed into a suitable
triangular form $\Lambda(t,\xi) + \mathcal{N}(t,\xi)$, where $\Lambda(t,\xi)$
is the diagonal matrix $diag(\lambda_{1}(t,\xi) \ldots \lambda_{m}(t,\xi))$ and
$\mathcal{N}(t,\xi)$ is a nilpotent upper triangular matrix. Hence, the global
hypoellipticity of $P$ is studied by analyzing the behavior of the eigenvalues
$\lambda_{j}(t,\xi)$ and its averages $\lambda_{0,j}(\xi)$, as $|\xi| \to
\infty$.
