Let $X$ be a complex manifold, let $E \to X$ be a holomorphic vector bundle of rank $r$, and let $h$ be a smooth Hermitian metric on $E$, with the convention that $h$ is conjugate-linear in the first argument and linear in the second. Let $U \subset X$ be an [open set](/page/Open%20Set) on which $e = (e_1,\dots,e_r)$ is a local holomorphic frame for $E$. Define the Hermitian metric matrix

\begin{align*} H: U \to \operatorname{Herm}_r^+(\mathbb{C}), \qquad H_{ij} = h(e_i,e_j). \end{align*}

Let $\nabla^E$ be the Chern connection of $(E,h)$, characterized by $(\nabla^E)^{0,1} = \bar{\partial}_E$ and $\nabla^E h = 0$. If $\theta \in \Omega^{1,0}(U;\operatorname{Mat}_r(\mathbb{C}))$ is the connection matrix in the row-frame convention

\begin{align*} \nabla^E e = e\theta, \end{align*}

equivalently $\nabla^E e_j = \sum_{i=1}^r e_i\theta_{ij}$, then

\begin{align*} \theta = H^{-1}\partial H. \end{align*}

Moreover, if $\Theta_E \in \Omega^{1,1}(U;\operatorname{Mat}_r(\mathbb{C}))$ is the curvature matrix defined by

\begin{align*} (\nabla^E)^2 e = e\Theta_E, \end{align*}

then

\begin{align*} \Theta_E = \bar{\partial}(H^{-1}\partial H). \end{align*}

Here $\partial$ and $\bar{\partial}$ act entrywise on matrix-valued forms, and matrix products of differential forms are taken with wedge product in the form component.