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Suppose that I have two matrices $A$ and $B$ which are both symmetric: $A=A^T, B=B^T$. Moreover, I know how to diagonalize both $A$ and $B$.

Now I would like to define $T=A^{1/2}BA^{1/2}$, which is also a symmetric matrix (in general, the commutator $[A,B]$ does not vanish). I then want to compute $\text{Tr}\left[\left(A^{1/2}BA^{1/2}\right)^n\right]$ for a positive integer $n$. My question is, can I use my knowledge of the diagonal form and/or eigenbasis of $A$ and $B$ to efficiently compute this trace? I think it isn't easy to compute the eigenvalues of $T$ with the available data so maybe this trace is also non-trivial to compute...

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    $\begingroup$ You can write down examples showing that the eigenvalues of $T$ are not determined by the eigenvalues of $A$ and $B$. This is a difficult computation in general; for example the computation of the partition function of the 2d Ising model has this form, where $T$ is the transfer matrix and it's easy to write down the eigenvalues and eigenvectors of $A$ and $B$ but the computation of the eigenvalues and eigenvectors of $T$ is quite a bit more involved. $\endgroup$ Commented Aug 5 at 0:41

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The trace is invariant under cyclic permutations, so the question is the same as computing the trace of $\text{Tr}\left(\left(AB\right)^n\right)$.

However, even in the case $n=1$ , this is not a trivial task unless you assume that $A$ and $B$ commute. Nevertheless, using the Cauchy–Schwarz inequality, you can obtain an upper bound.

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