  In this paper, we propose linear maps over the space of all polynomials
$f(x)$ in $\mathbb{F}_q[x]$ that map $0$ to itself, through their evaluation
map. Properties of these linear maps throw up interesting connections with
permutation polynomials. We study certain properties of these linear maps. We
propose to classify permutation polynomials by identifying the generalized
eigenspaces of these maps, where the permutation polynomials reside. As it
turns out, several classes of permutation polynomials studied in literature
neatly fall into classes defined using these linear maps. We characterize the
shapes of permutation polynomials that appear in the various generalized
eigenspaces of these linear maps. For the case of $\mathbb{F}_p$, these
generalized eigenspaces provide a degree-wise distribution of polynomials (and
therefore permutation polynomials) over $\mathbb{F}_p$.
  We show that for $\mathbb{F}_q$, it is sufficient to consider only a few of
these linear maps. The intersection of the generalized eigenspaces of these
linear maps contain (permutation) polynomials of certain shapes. In this
context, we study a class of permutation polynomials over $\mathbb{F}_{p^2}$.
We show that the permutation polynomials in this class are closed under
compositional inverses. We also do some enumeration of permutation polynomials
of certain shapes.
