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By Cayley transformation, there is an interplay between matrix-valued Carathéodory and Schur functions in the unit disk. One of the main aims of the paper is to study this interplay with a view to the fact that the matrix function given by the values of a Carathéodory function via Moore–Penrose inverses is also a Carathéodory function. In doing so, we also analyze ranges and null spaces of the values...