TITLE: If $A$ is diagonal and positive and $B$ is skew-hermitian, does $AB$ have only pure imaginary eigenvalues?
QUESTION [5 upvotes]: Let $A$ be diagonal with strictly positive (real) entries, and let $B$ be skew hermitian.  Can it be shown that the eigenvalues of $AB$ are pure imaginary?
I suspect this also holds in the more general case that $A$ is symmetric positive definite.

REPLY [3 votes]: Your intuition is right. When $A$ is positive definite and $B$ is skew-Hermitian, $AB$ is similar to $A^{1/2}BA^{1/2}$ (as $A^{1/2}BA^{1/2}=A^{-1/2}(AB)A^{1/2}$), which is skew-Hermitian. Hence it has a purely imaginary spectrum.