Editing Trace (linear algebra) (section)

=== Cyclic property ===
More generally, the trace is ''invariant under [[circular shift]]s'', that is,

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|equation = <math>\operatorname{tr}(\mathbf{A}\mathbf{B}\mathbf{C}\mathbf{D}) = \operatorname{tr}(\mathbf{B}\mathbf{C}\mathbf{D}\mathbf{A}) = \operatorname{tr}(\mathbf{C}\mathbf{D}\mathbf{A}\mathbf{B}) = \operatorname{tr}(\mathbf{D}\mathbf{A}\mathbf{B}\mathbf{C}).</math>
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This is known as the ''cyclic property''.

Arbitrary permutations are not allowed: in general,
<math display="block">\operatorname{tr}(\mathbf{A}\mathbf{B}\mathbf{C}\mathbf{D}) \ne \operatorname{tr}(\mathbf{A}\mathbf{C}\mathbf{B}\mathbf{D}) ~.</math>

However, if products of ''three'' [[symmetric matrix|symmetric]] matrices are considered, any permutation is allowed, since:
<math display="block">\operatorname{tr}(\mathbf{A}\mathbf{B}\mathbf{C}) = \operatorname{tr}\left(\left(\mathbf{A}\mathbf{B}\mathbf{C}\right)^{\mathsf T}\right) = \operatorname{tr}(\mathbf{C}\mathbf{B}\mathbf{A}) = \operatorname{tr}(\mathbf{A}\mathbf{C}\mathbf{B}),</math>
where the first equality is because the traces of a matrix and its transpose are equal. Note that this is not true in general for more than three factors.