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A holomorphic matrix function on a domain $D\subseteq \mathbb{C}$ is a matrix-valued function $A(z)$ whose entries are holomorphic functions. Differentiation is entrywise, and products satisfy the ordered product rule $(AB{)}^{\prime }=A^{\prime }B+A{B}^{\prime }$. The order matters because matrix multiplication need not be commutative.

If $A(z)$ is square and $\det A(z)\ne 0$ throughout $D$, the adjugate formula shows its inverse is also holomorphic. Differentiating $A{A}^{-1}=I$ yieldsFor $A(z)=\left(\begin{array}{cc}1& z\\ 0& 1\end{array}\right)$, its inverse is $\left(\begin{array}{cc}1& -z\\ 0& 1\end{array}\right)$ and the identity is immediate. For a fixed matrix $B$, the resolvent $(zI-B{)}^{-1}$ is holomorphic off its spectrum. This links complex analysis with eigenvalue problems; it is distinct from applying a scalar function to a matrix by functional calculus.

A jump matrix is a prescribed invertible matrix-valued function $J$ on an oriented contour in a matrix Riemann-Hilbert problem. With plus boundary on the left of the contour, the unknown holomorphic matrix function satisfiesMatrix multiplication order matters. Where $M_{-}$ is invertible, $J=M_{-}^{-1}{M}_{+}$, rather than $M_{+}{M}_{-}^{-1}$ in general. Reversing contour orientation swaps the sides and replaces $J$ by $J^{-1}$.

For the positively oriented unit circle, a constant invertible matrix $J=A$ is realized by $M=A$ inside and $M=I$ outside. The exterior solution is normalized to $I$ at infinity. More complicated jumps encode boundary values of spectral or scattering data; prescribed poles, endpoint behavior and normalization must also be specified to define the full Riemann-Hilbert problem.^[ref]