= Holomorphic matrix function
{title2=$A(z)$}
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> $(AB)'=A'B+AB'$. The order matters because <matrix multiplication> need not be <commutative>.
If $A(z)$ is square and $\det A(z)\ne0$ throughout $D$, the adjugate formula shows its inverse is also <holomorphic>. Differentiating $AA^{-1}=I$ yields
$$
(A^{-1})'=-A^{-1}A'A^{-1}.
$$
For $A(z)=\begin{pmatrix}1&z\\0&1\end{pmatrix}$, its inverse is $\begin{pmatrix}1&-z\\0&1\end{pmatrix}$ 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.
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