Which of the following statements is true ?
Trace (AB) = Trace (BA)
The trace of a square matrix is the sum of its diagonal elements, \(\mathrm{Tr}\,M=\sum_{i}M_{ii}\).
For a product, \(\mathrm{Tr}(AB)=\sum_{i}\sum_{j}A_{ij}B_{ji}\). Renaming the dummy indices gives \(\sum_{j}\sum_{i}B_{ji}A_{ij}=\mathrm{Tr}(BA)\), so \(\mathrm{Tr}(AB)=\mathrm{Tr}(BA)\) for all square matrices of the same order, whether or not they commute.
This result only allows cyclic permutations: \(\mathrm{Tr}(ABC)=\mathrm{Tr}(BCA)=\mathrm{Tr}(CAB)\). Swapping just two factors, as in \(\mathrm{Tr}(BAC)\) or \(\mathrm{Tr}(BACD)\), is not a cyclic rearrangement and fails in general for non-commuting matrices.
The trace is linear but not multiplicative: for example with \(A=B=I_{2}\), \(\mathrm{Tr}(AB)=2\) while \(\mathrm{Tr}\,A\cdot\mathrm{Tr}\,B=4\).
So the only identity valid for arbitrary matrices is the equality of the traces of the two orderings of a product of two matrices.
Hence, Trace (AB) = Trace (BA).
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.................... is a client server application that identifies each host on the internet with a unique user-friendly name.
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(i) Kernel
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(iii) Nucleus
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