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I've been thinking about this for a while and I've found that for the 2x2 case the matrices are just $\{e_{11}, e_{12} + e_{21}, e_{21}, e_{22}\}$ where $e_{ij}$ is the matrix with $1$ on the $ij$ entry, $0$'s everywhere else. However I can't seem to see any patterns with higher orders. I need to find the basis for the $n \times n$ case.

Any help is appreciated.

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    Typeset everything in $\LaTeX$. Let me know if I messed anything up.2011-12-02
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    You really want to define it as $ \mbox{tr}(A B^T).$ If you want the identity matrix to give $H(I,I) = 1$ you take $\frac{1}{n} \; \mbox{tr}(A B^T).$2011-12-02
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    Well no, the bilinear form I'm given is defined as H(A,B) = tr(AB), not tr(AB^t) unless I misunderstood what you meant.2011-12-02
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    Note that $H(e_{21},e_{21})=0.$2011-12-02

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