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## Main Question or Discussion Point

I have two real symmetric matrices [itex]A[/itex] and [itex]B[/itex] with the following additional properties. I would like to know how the eigenvalues of the product [itex]AB[/itex], is related to those of [itex]A[/itex] and [itex]B[/itex]? In particular what is [itex]\mathrm{trace}(AB)[/itex]?

[itex]A[/itex] contains only 0s on its diagonal. Off diagonal terms are either 0 or 1.

[itex]B[/itex] also contains only 0s on its diagonal. Its off diagonal terms are positive real numbers.

If equalities don't exist, some bounds would also be helpful.

Thanks.

[itex]A[/itex] contains only 0s on its diagonal. Off diagonal terms are either 0 or 1.

[itex]B[/itex] also contains only 0s on its diagonal. Its off diagonal terms are positive real numbers.

If equalities don't exist, some bounds would also be helpful.

Thanks.