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I was wondering for what matrices (over $\mathbb{C}$) $A$ and $B$ is the equation $\operatorname{tr}(AB) = \operatorname{tr}(A)\operatorname{tr}(B)$ satisfied?

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    Even when $B=I$ and dimension >1 there is no way2012-04-24

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Even when $A, B$ are diagonal matrices, the equality is less likely to hold, let alone general matrices.