According to Yau’s proof of Calabi conjecture:
If we choose another coordinate system so that $g_{i\bar{j}}=\delta_{ij}$ and $\varphi_{i\bar{j}}=\delta_{ij}\varphi_{i\bar{i}}$, then we have ……
Here $g_{i\bar{j}}$ is the coefficients of hermitian metric, $\varphi$ is a real valued smooth function, in local coordinate $\varphi_{i\bar{j}}=\frac{\partial^2 \varphi}{\partial z^i\partial\bar{z}^j}$, thus $(g_{i\bar{j}})$ and $(\varphi_{i\bar{j}})$ are both hermitian matrices. The diagonalization of a matrix $A$ is in the sense of $QA\bar{Q}^T, \text{det}(Q)\neq0$, (I believe this way of diagonalization has its own name, maybe hermitian diagonalize?) in order to be consistent with change in holomorphic coordinate system.
In short, I don’t understand how to hermitian diagonalize two hermitian matrices, one to the identity, another to the diagonal of its own.
Thanks!