Useful matrix inner products

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What are some interesting/useful examples of matrix inner products on square complex matrices, other than $\langle A, B \rangle = Tr(A^{\dagger}B)$?

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It's hard to find anything as useful as the one you mention, but this might be of interest. https://mathoverflow.net/questions/115192/power-log-distance-between-matrices/115227#115227 (ps. again I would have made this a comment rather than an answer, but I still can't do that)