The anti-symmetrized sum of traces of products of \(k\) matrices is always 0 for even \(\mathrm{k}\). But it's generally nonzero for odd \(\mathrm{k}\), which turns out to be a useful quantity in certain applications. #math #algebra #matrices

Let \(A_{1}, A_{2}, \ldots, A_{k}\) be \(n \times n\) matrices. Show that the antisymmetrized sum of traces

\[

B=\sum_{\pi \in S_{k}} \operatorname{sgn}(\pi) \operatorname{Tr}\left[A_{\pi(1)} A_{\pi(2)} \cdots A_{\pi(k)}\right]

\]

satisfies

\[

B=(-1)^{k-1} B

\]

It then follows that \(B=0\) for even \(k\) (for all \(A_{i}\) 's!). Show, however, that \(B\) can be nonzero for odd \(k\).

The sum runs over all permutations \(\pi\) of the indices \(\{1,2, \ldots, k\}\). The sign of \(\pi, \operatorname{sgn}(\pi)\), is \(+1\) if \(\pi\) is a product of an even number of transpositions, and is \(-1\) if \(\pi\) is a product of an odd number.