I HATED this exercise
The trace is invariant under cyclic permutations of A,B,C. Exercise: Prove this result for the more general case of n matrices 𝐴₁, 𝐴₂, ... 𝐴𝑛.pic.twitter.com/h5cPoKzsmD
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it's like 2 lines! haha
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Generic (Fubini's theorem for measures μ{1} ... μ{n}): ∫...∫dμ{1}(a[1])...dμ{n}(a[n]){f1(a[1],a[2])...fn(a[n],a[1])} = ∫...∫dμ{n}(a[n])dμ{1}(a[1])...dμ{n-1}(a[n-1]){fn(a[n],a[1])f1(a[1],a[2])...fn-1(a[n-1],a[n])}.
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I’d take the scenic route to the QED using strong induction
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Suggestion use graphs of the tensor network notation: Tr(ABC)=A_ijB_jkC_ki=B_jkC_kiA_ij= B_ijC_jkA_ki=Tr(BCA) etc. Since i,j and k are dummies. The graph of Tr(ABC) has no external lines so it is a scalar.pic.twitter.com/tJQ6b8M3Q5
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