$\newcommand{\diag}{\operatorname{diag}}$I'm confused about converting between explicit matrix expressions and index notation. Take $\eta_{\mu\nu}=\diag(-1,1,1,1)$ and $A$ some $4\times4$ matrix. For example,
\begin{align} & (\diag(-1,1,1,1).A.\diag(-1,1,1,1))^T\\[8pt] = {} & (\eta_{\mu\rho} A^{\rho\sigma} \eta_{\sigma\nu})^T \\[8pt] = {} & \eta_{\nu\sigma} A^{\sigma\rho} \eta_{\rho\mu} \\[8pt] = {} & \diag(-1,1,1,1).A.\diag(-1,1,1,1) \end{align} but it's not its own transpose! Which step is wrong? Also, $$A^\mu{}_\mu=A^{\mu\rho}\eta_{\rho\mu}=A.\diag(-1,1,1,1)$$ but it should be the trace of $A$. How shall we convert between index notation and explicit matrix notation such that every step is mathematically justified and no confusions arise?