223 Appendix F: Heavy Quark Production Radiation Derivation it follows that G can be chosen to be symmetric without loss of generality. It can therefore be diagonalized by a unitary transformation. Define the diago- nalized metric as G = UGU − $^{1}$, (F.3) det G = det( UGU − $^{1}$) = det( U − $^{1}$UG ) = det G. (F.4) ∂$_{µ}$g = ∂$_{µ}$e ln det $^{G}$. (F.5) where U is the unitary matrix whose rows are the (normalized) eigenvectors of G . Note that Consider Since det G = det G , from above, and, trivially from the definition, ln det G = Tr ln G , we have that (where G$_{i}$ is the i th eigenvalue) ∂$_{µ}$g = g∂$_{µ}$ Tr ln G = g∂$_{µ}$ ∑ i ln G$_{i}$ = g ∑ i G − 1 i ∂$_{µ}$ G$_{i}$ = g Tr G − $^{1}$∂$_{µ}$ G . (F.6)