Let $(E,h)$ be a holomorphic Hermitian vector bundle with local frame $\{e_i\}$. Denote by $H$ the matrix of $h$ in $\{e_i\}$. I found two different expression for the Chern connection 1-form $\omega$:
In some place it says that $\omega=H^{-1}\partial H$ and in other it says $\omega=\partial H\cdot H^{-1}$. Are they actually the same? if not is there a mistake? if yes which one is the true one?
First expression: https://ncatlab.org/nlab/files/ChernConnections.pdf
Second expression: https://www.math.uh.edu/~shanyuji/Complex/Geom/cx-26.pdf