Holomorphic matrix function ($A(z)$)

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 $(AB{)}^{\prime }=A^{\prime }B+A{B}^{\prime }$. 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.