  Given a purely contractive matrix-valued analytic function $\Theta$ on the
unit disc $\bm{D}$, we study the $\mc{U} (n)$-parameter family of unitary
perturbations of the operator $Z_\Theta$ of multiplication by $z$ in the
Hilbert space $L^2_\Theta$ of $n-$component vector-valued functions on the unit
circle $\bm{T}$ which are square integrable with respect to the matrix-valued
measure $\Om_\Theta$ determined uniquely by $\Theta$ and the matrix-valued
Herglotz representation theorem.
  In the case where $\Theta $ is an extreme point of the unit ball of bounded
$\bm{M}_n$-valued functions we verify that the $\mc{U} (n)$-parameter family of
unitary perturbations of $Z_\Theta ^*$ is unitarily equivalent to a $\mc{U}
(n)$-parameter family of unitary perturbations of $X_\Theta$, the restriction
of the backwards shift in $H^2_n (\bm{D})$, the Hardy space of $\bm{C} ^n$
valued functions on the unit disc, to $K^2_\Theta$, the de Branges-Rovnyak
space constructed using $\Theta$. These perturbations are higher dimensional
analogues of the unitary perturbations introduced by D.N. Clark in the case
where $\Theta$ is a scalar-valued ($n=1$) inner function, and studied by E.
Fricain in the case where $\Theta$ is scalar-valued and an extreme point of the
unit ball of $H^\infty (\bm{D})$...
  A matrix-valued disintegration theorem for the Aleksandrov-Clark measures
associated with matrix-valued contractive analytic functions $\Theta$ is
obtained as a consequence of the Weyl integration formula for $\mc{U}(n)$
applied to the family of unitary perturbations of $Z_\Theta$...
