Let $(E,h) \to X$ be a Hermitian holomorphic vector bundle of rank $r$ over a complex manifold $X$. Let $U \subset X$ be an [open set](/page/Open%20Set) admitting a holomorphic frame $e = (e_1,\dots,e_r)$, and let $H: U \to \operatorname{Herm}_r^{+}$ be the Hermitian metric matrix in this frame, with entries $H_{ij} = h(e_j,e_i)$. Write sections over $U$ as column-vector-valued functions relative to $e$, and use the convention that a connection matrix acts by left multiplication: \begin{align*} D = d + A. \end{align*} For the Chern connection $D$, the local connection matrix is \begin{align*} A = H^{-1}\partial H. \end{align*} If the curvature is defined by $F_D = D^2$, then the curvature matrix in the frame $e$ is \begin{align*} F_D = \bar{\partial}(H^{-1}\partial H). \end{align*}