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Suppose that $A_1,\dots,A_n$ are square, symmetric, positive semi-definite matrices. Consider the product $A_1 \cdot A_2 \cdots A_n$. I am aware that it is not necessarily symmetric.

I am wondering, however, if anything can be said about its eigenvalues. Can it be shown that its eigenvalues are real? Can it be shown that its eigenvalues are non-negative?

I am only aware about results for $n=2$, see this link for example. It would be nice if results here which says that for symmetric positive definite matrices $A,B$, the eigenvalues of $AB$ are bounded below by the product of eigenvalues of $A,B$, could be extended to the case when there are more than 2 factors in the product and each factor is symmetric positive semi-definite.

secondrate
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    you'd be wise to focus on complex hermitian matrices $A,B,C$ and observe "On the other hand, there is no reason why expressions such as ${\hbox{tr}(e^A e^B e^C)}$ need to be positive or even real, so the obvious extension of the Golden-Thompson inequality to three or more Hermitian matrices fails." qtd from here: https://terrytao.wordpress.com/2010/07/15/the-golden-thompson-inequality/ – user8675309 Nov 10 '20 at 18:48

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