  This paper deals with families of matrix-valued Aleksandrov--Clark measures
$\{\boldsymbol{\mu}^\alpha\}_{\alpha\in\mathcal{U}(n)}$, corresponding to
purely contractive $n\times n$ matrix functions $b$ on the unit disc of the
complex plane. We do not make other apriori assumptions on $b$. In particular,
$b$ may be non-inner and/or non-extreme. The study of such families is mainly
motivated from applications to unitary finite rank perturbation theory.
  A description of the absolutely continuous parts of $\boldsymbol{\mu}^\alpha$
is a rather straightforward generalization of the well-known results for the
scalar case ($n=1$).
  The results and proofs for the singular parts of matrix-valued
$\boldsymbol{\mu}^\alpha$ are more complicated than in the scalar case, and
constitute the main focus of this paper. We discuss matrix-valued
Aronszajn--Donoghue theory concerning the singular parts of the Clark measures,
as well as Carath\'{e}odory angular derivatives of matrix-valued functions and
their connections with atoms of $\boldsymbol{\mu}^\alpha$. These results are
far from being straightforward extensions from the scalar case: new phenomena
specific to the matrix-valued case appear here. New ideas, including the notion
of directionality, are required in statements and proofs.
