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This paper deals with families of matrix-valued Aleksandrov--Clark measures$\{\boldsymbol{\mu}^\alpha\}_{\alpha\in\mathcal{U}(n)}$, corresponding topurely contractive $n\times n$ matrix functions $b$ on the unit disc of thecomplex 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 mainlymotivated 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 thescalar 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, andconstitute the main focus of this paper. We discuss matrix-valuedAronszajn--Donoghue theory concerning the singular parts of the Clark measures,as well as Carath\'{e}odory angular derivatives of matrix-valued functions andtheir connections with atoms of $\boldsymbol{\mu}^\alpha$. These results arefar from being straightforward extensions from the scalar case: new phenomenaspecific to the matrix-valued case appear here. New ideas, including the notionof directionality, are required in statements and proofs.

Matrix-valued Aleksandrov--Clark measures and Carath\'{e}odory angular derivatives