How can I write the determinant as traces in this paper on complex matrices?

[rbwang1225]

I am reading the paper http://arxiv.org/abs/hep-th/9701037.
In equation (2), the author write the determinat as traces, but I don't know how to do this.
I know that $det(e^A)=e^{tr(A)}$, where $A$ is a complex matrix, and $det(A)=e^{tr(L)}$, where $e^L=A$ and $L$ is also a complex matrix.
The problem becomes how to find the logrithm of the matrix $g_{mn}+i$\tilde{F}_{mn}.
Above is what I can figure out now.

Any help would be vary appreciated!
Regards.

 

[scottbekerham]

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