For a square matrix A and a function f, holomorphic on a neighbourhood of the
spectrum of the matrix A, we can define the matrix f(A). In this thesis, we present
the definition via the resolvent, i. e. the inverse of the matrix zI - A, which is a
matrix of holomorphic functions on the complement of the spectrum of the matrix A,
and we show that matrix functions defined in this way behave well under addition,
multiplication and composition of functions. We then seek necessary and sufficient
conditions for a given holomorphic function to induce a surjective mapping on the
appropriate subset of square matrices, and we explore under what conditions a
matrix of polynomials or entire functions has a well-defined logarithm.
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