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What is meant by a cyclic rotation of a matrix, specifically in proving that j*k-k*j=i*l where j,k,l are cyclic rotations of the Pauli spin matrices sigma x, sigma y, and sigma z

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By the term a cyclic rotation your source means a cyclic permutation. –  Qmechanic Dec 11 '12 at 17:41
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The statement "A,B,C" are cyclic rotations (more often: cyclic permutations) of "E,F,G" means that "A,B,C" is either "E,F,G" or "F,G,E" or "G,E,F", in one of these three orders (but not in the remaining orders "E,G,F", "F,E,G", "G,F,E"). Relatively to "E,F,G", these three letters were ordered by one of the three elements of the so-called cyclic group ${\mathbb Z}_3$.

Similarly for cyclic rotations of $k$ elements and the group ${\mathbb Z}_k$.

In your example, if you used a non-cyclic permutation of the Pauli matrices, you would get $j*k-k*j = -i*l$ with the minus sign on the right hand side because the Pauli matrices anticommute.

In particular, let me stress that no operation is applied "inside" the matrices. They're treated as wholes, as elements of a set, that are being permuted with other elements.

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