Consider a Riemann surface $\Sigma$ and a holomorphic vector bundle $\mathcal{E} \to \Sigma$ of rank $N$.
I just came across the remark that in order to endow $\mathcal{E}$ with a Hermitian metric it is necessary that the positive definite Hermitian matrices of rank $N$ constitute a convex set in GL($N,\mathbb{C})$.
I would like to understand why this is so, but I have trouble finding the right source where I can read about this - any hints as to why I need the convexity, or reference suggestions targeted to this aspect of complex manifold theory in case this question is too broad, would be very helpfull! many thanks !